Systems and methods for treatment deconvolution using multi-scale kernels

ABSTRACT

Deconvolution systems and methods based on cornea smoothing can be used to obtain an ablation target or treatment shape that does not induce significant high order aberrations such as spherical aberration. Exemplary ablation targets or treatment shapes can provide a post-operative spherical aberration that is equal to or below a naturally occurring amount of spherical aberration.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. Non provisional applicationSer. No. 14/523,467, filed on Oct. 24, 2014, which is a continuation ofU.S. Non provisional application Ser. No. 14/453,068, filed on Aug. 6,2014, which claims the benefit of priority to U.S. ProvisionalApplication No. 61/871,120, filed Aug. 28, 2013. This application isrelated to U.S. Patent Application No. 61/708,815 filed Oct. 2, 2012.This application is also related to U.S. Pat. No. 7,926,490 issued Apr.19, 2011, and U.S. patent application Ser. No. 13/554,276, filed Jul.20, 2012. The entire content of each of the above filings is herebyincorporated by reference for all purposes.

BACKGROUND OF THE INVENTION

Embodiments of the present invention related to the field of visiontreatment, and in particular to systems and methods for generating ormodifying optical treatment shapes.

The post-operative induction of high-order aberrations (HOAs),especially spherical aberration (SA), remains an important issue forlaser vision correction technology.

It has been found that post-operative cornea remodeling is a significantroot cause of SA induction. One main effect of the cornea remodelinginvolves the smoothing of epithelium at the anterior surface of the eye,where the epithelium tends to grow thicker and fill in the dips of thecornea surface as created by refractive surgery. Epithelial smoothingcan result in regression following refractive surgery, and sometimesleads to induced high-order aberrations that are particularly strong forhigh myopia and hyperopia cases.

Certain techniques have been proposed for minimizing inducedpost-operative SA, including linear adjustment of the basis data andnomogram adjustments. Although such techniques can provide benefits topatients in need thereof, further improvements would be desirable.Embodiments of the present invention provide solutions to address theseand other outstanding needs.

BRIEF SUMMARY OF THE INVENTION

It has been discovered that deconvolution techniques based on a corneasmoothing model can be used to obtain an ablation target or treatmentshape that induces little or no post-operative SA. In some instances,these ablation targets or treatment shapes can provide a post-operativeSA that is equal to or below a naturally occurring amount of SA.

Hence, embodiments of the present invention encompass systems andmethods for obtaining a modified ablation target that is capable ofeliminating, reducing, or minimizing a systematic trend inpost-operatively induced spherical aberration. In some cases, themodification of the target shape introduces only a small increase in therequired depth for the ablation. Hence, such techniques are helpful inproviding safe and effective treatments. In some cases, the modificationof the target shape may change the peripheral cornea profile, which canaffect the SA without changing the central refractive power.

In some instances, embodiments encompass techniques for determining avision treatment for an eye of a patient, which may include obtaining anoriginal target profile for the eye of the patient, obtaining a spatialdomain kernel filter (e.g. based on an inverse Fourier transform of aFourier domain noise filter), convolving the original target profilewith the spatial domain kernel filter, and determining the visiontreatment based on the convolved profile.

Embodiments of the present invention can be readily adapted for use withexisting laser systems and other optical treatment devices. Althoughsystem, software, and method embodiments of the present invention aredescribed primarily in the context of a laser eye surgery system, itshould be understood that embodiments of the present invention may beadapted for use in or in combination with alternative eye treatmentprocedures, systems, or modalities, such as spectacle lenses,intraocular lenses, accommodating IOLs, contact lenses, corneal ringimplants, collagenous corneal tissue thermal remodeling, corneal inlays,corneal onlays, other corneal implants or grafts, and the like.Relatedly, systems, software, and methods according to embodiments ofthe present invention are well suited for customizing any of thesetreatment modalities to a specific patient. Thus, for example,embodiments encompass custom preformed lenses, intraocular lenses,custom contact lenses, custom corneal implants, and the like, which canbe configured to treat or ameliorate any of a variety of visionconditions in a particular patient based on their unique ocularcharacteristics or anatomy. Additionally, the modified ablation targetor target shape may be implemented via other non-ablative lasertherapies, such as laser-incised custom lenticule shapes and subsequentextraction and laser-based corneal incision patterns.

In some instances, these techniques can be carried out in conjunctionwith treatments provided by any of a variety of laser devices, includingwithout limitation the WaveScan® System and the STAR S4® Excimer LaserSystem both by Abbott Medical Optics Inc., the WaveLight® AllegrettoWave® Eye-Q laser, the Schwind Amaris™ lasers, the 217P excimerworkstation by Technolas PerfectVision GmbH, the Mel 80™ laser by CarlZeiss Meditec, Inc., and the like.

In one aspect, embodiments of the present invention encompass systemsand methods for determining a vision treatment for an eye of a patient.Exemplary techniques may include, for example, receiving, at an input,an original target profile for the eye of the patient, and convolvingthe original target profile with the spatial domain kernel filter. Thespatial domain kernel filter can be based on an inverse Fouriertransform of a Fourier domain noise filter. Techniques may also includedetermining the vision treatment based on the convolved profile.Optionally, techniques may include administering the treatment to thepatient. In some instances, the Fourier domain noise filter is based ona conjugate of a Fourier domain complex matrix. In some instances, theFourier domain noise filter is based on a modulus of a Fourier domaincomplex matrix. In some instances, the Fourier domain noise filter isbased on a conjugate of a Fourier domain complex matrix and a modulus ofthe Fourier domain complex matrix. According to some embodiments, theFourier domain noise filter is characterized by fraction having anumerator comprising a conjugate of a Fourier domain complex matrix anda denominator comprising a modulus of the Fourier domain complex matrix.In some cases, the Fourier domain complex matrix is characterized by theformula

${K( {k_{x},k_{y}} )} = \frac{1}{1 + \frac{\sigma^{2}( {k_{x}^{2} + k_{y}^{2}} )}{( {0.5\mspace{14mu}{dL}} )^{2}}}$where σ represents a diffusion coefficient, k_(x) and k_(y) representfrequency domain variables, and dL represents a mesh size. In somecases, σ has a value of 0.35 mm and dL has a value of 0.1 mm.Optionally, σ may have a value within a range from about 0.2 mm to about0.5 mm. In some cases, σ may have a value within a range from about 0.33mm to about 0.4 mm. Optionally, the denominator can be characterized bythe expression |K(k_(x), k_(y))|^(n), where n is an integer having avalue of 2 or more. In some instances, the denominator can becharacterized by the expression [|K(k_(x), k_(y))|^(n)+SNR²] where n isan integer having a value of 2 or more and SNR represents a signal tonoise ratio value. In some instances, the convolved profile includes atransition zone radius, and a method may further include zeroing theconvolved profile at locations outside of the transition zone radius. Insome instances, the original target profile may include an originalrefractive spherical equivalent value within a 4 mm diameter area, andthe convolved target profile may include a target refractive sphericalequivalent value within a 4 mm diameter area. Optionally, the method mayfurther include scaling the original refractive spherical equivalentwith the target refractive spherical equivalent value. Some methods mayalso include elevating the convolved profile so that a lowest point onthe convolved profile is zero or greater. In some instances, a convolvedprofile includes a transition zone radius, and methods may includeapplying a damping multiplier at or near the transition zone radius. Insome instances, the target shape includes an optical zone having aperiphery, and the convolution effects a change in the target shape nearthe periphery of the optical zone.

In another aspect, embodiments of the present invention encompasssystems for determining a vision treatment for an eye of a patient.Exemplary systems may include an input that receives an original targetprofile for the eye of the patient, and a convolution module thatconvolves the original target profile with a spatial domain kernelfilter. The spatial domain kernel filter can be based on an inverseFourier transform of a Fourier domain noise filter. Systems may alsoinclude a treatment generation or determination module that determinesthe vision treatment based on the convolved profile. Optionally, systemscan include a treatment delivery module that delivers the treatment tothe patient. In some instances, the Fourier domain noise filter is basedon a conjugate of a Fourier domain complex matrix. In some instances,the Fourier domain noise filter is based on a modulus of a Fourierdomain complex matrix. In some instances, the Fourier domain noisefilter is based on a conjugate of a Fourier domain complex matrix and amodulus of the Fourier domain complex matrix. According to someembodiments, the Fourier domain noise filter is characterized byfraction having a numerator comprising a conjugate of a Fourier domaincomplex matrix and a denominator comprising a modulus of the Fourierdomain complex matrix. In some cases, the Fourier domain complex matrixis characterized by the formula

${K( {k_{x},k_{y}} )} = \frac{1}{1 + \frac{\sigma^{2}( {k_{x}^{2} + k_{y}^{2}} )}{( {0.5\mspace{14mu}{dL}} )^{2}}}$where σ represents a diffusion coefficient, k_(x) and k_(y) representfrequency domain variables, and dL represents a mesh size. In somecases, σ has a value of 0.35 mm and dL has a value of 0.1 mm.Optionally, σ may have a value within a range from about 0.2 mm to about0.5 mm. In some cases, σ may have a value within a range from about 0.33mm to about 0.4 mm. Optionally, the denominator can be characterized bythe expression |K(k_(x), k_(y))|^(n), where n is an integer having avalue of 2 or more. In some instances, the denominator can becharacterized by the expression [|K(k_(x), k_(y))|^(n)+SNR²] where n isan integer having a value of 2 or more and SNR represents a signal tonoise ratio value. In some instances, the convolved profile includes atransition zone radius, and the convolution module can zero theconvolved profile at locations outside of the transition zone radius. Insome instances, the original target profile may include an originalrefractive spherical equivalent value within a 4 mm diameter area, andthe convolved target profile may include a target refractive sphericalequivalent value within a 4 mm diameter area. Optionally, theconvolution module can scale the original refractive sphericalequivalent with the target refractive spherical equivalent value. Insome cases, the convolution module can elevate the convolved profile sothat a lowest point on the convolved profile is zero or greater. In someinstances, a convolved profile includes a transition zone radius, andthe convolution module can apply a damping multiplier at or near thetransition zone radius. In some instances, the target shape includes anoptical zone having a periphery, and the convolution module can effect achange in the target shape near the periphery of the optical zone.

In still another aspect, embodiments of the present invention encompasscomputer program products for determining a vision treatment for an eyeof a patient. An exemplary computer program product may be embodied on anon-transitory tangible computer readable medium, and may includecomputer code for receiving an original target profile for the eye ofthe patient, computer code for convolving the original target profilewith a spatial domain kernel filter, and computer code for determiningthe vision treatment based on the convolved profile. The spatial domainkernel filter may be based on an inverse Fourier transform of a Fourierdomain noise filter. Optionally, computer program products may includecomputer code for delivering or administering the treatment to thepatient. In some instances, the Fourier domain noise filter is based ona conjugate of a Fourier domain complex matrix. In some instances, theFourier domain noise filter is based on a modulus of a Fourier domaincomplex matrix. In some instances, the Fourier domain noise filter isbased on a conjugate of a Fourier domain complex matrix and a modulus ofthe Fourier domain complex matrix. According to some embodiments, theFourier domain noise filter is characterized by fraction having anumerator comprising a conjugate of a Fourier domain complex matrix anda denominator comprising a modulus of the Fourier domain complex matrix.In some cases, the Fourier domain complex matrix is characterized by theformula

${K( {k_{x},k_{y}} )} = \frac{1}{1 + \frac{\sigma^{2}( {k_{x}^{2} + k_{y}^{2}} )}{( {0.5\mspace{14mu}{dL}} )^{2}}}$where σ represents a diffusion coefficient, k_(x) and k_(y) representfrequency domain variables, and dL represents a mesh size. In somecases, σ has a value of 0.35 mm and dL has a value of 0.1 mm.Optionally, σ may have a value within a range from about 0.2 mm to about0.5 mm. In some cases, σ may have a value within a range from about 0.33mm to about 0.4 mm. Optionally, the denominator can be characterized bythe expression |K(k_(x), k_(y))|^(n), where n is an integer having avalue of 2 or more. In some instances, the denominator can becharacterized by the expression [|K(k_(x), k_(y))|^(n)+SNR²] where n isan integer having a value of 2 or more and SNR represents a signal tonoise ratio value. In some instances, the convolved profile includes atransition zone radius, and the computer code for convolving can includecomputer code for zeroing the convolved profile at locations outside ofthe transition zone radius. In some instances, the original targetprofile may include an original refractive spherical equivalent valuewithin a 4 mm diameter area, and the convolved target profile mayinclude a target refractive spherical equivalent value within a 4 mmdiameter area. Optionally, the computer code for convolving can includecomputer code for scaling the original refractive spherical equivalentwith the target refractive spherical equivalent value. In some cases,the computer code for convolving can include computer code for elevatingthe convolved profile so that a lowest point on the convolved profile iszero or greater. In some instances, a convolved profile includes atransition zone radius, and the computer code for convolving can includecomputer code for applying a damping multiplier at or near thetransition zone radius. In some instances, the target shape includes anoptical zone having a periphery, and the computer code for convolvingcan include computer code for effecting a change in the target shapenear the periphery of the optical zone.

For a fuller understanding of the nature and advantages of the presentinvention, reference should be had to the ensuing detailed descriptiontaken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a laser ablation system according to an embodiment ofthe present invention.

FIG. 2 illustrates a simplified computer system according to anembodiment of the present invention.

FIG. 3 illustrates a wavefront measurement system according to anembodiment of the present invention.

FIG. 3A illustrates another wavefront measurement system according toanother embodiment of the present invention.

FIG. 4 depicts aspects of a method for determining a vision treatmentfor an eye, according to embodiments of the present invention.

FIG. 5 depicts aspects of a method for modifying a target shapeaccording to embodiments of the present invention.

FIG. 6A shows post-operative values and FIG. 6B shows aspects of opticaland transition zones according to embodiments of the present invention.

FIG. 7 shows aspects of simulated epithelium thickness profilesaccording to embodiments of the present invention.

FIGS. 8A and 8B show aspects of flap SA and sigma relationshipsaccording to embodiments of the present invention.

FIGS. 9A to 9C depict aspects of post-operative SA and pre-operativeMRSE or SE relationships according to embodiments of the presentinvention.

FIGS. 10A and 10B illustrate aspects of spherical aberration errors fordeconvolution according to embodiments of the present invention.

FIGS. 11A and 11B show aspects of rescaling coefficients and refractionerrors, respectively, according to embodiments of the present invention.

FIGS. 12A and 12B depict aspects of effects of deconvolution on cylinderrefraction according to embodiments of the present invention.

FIGS. 12C and 12D illustrate aspects of ablation profile modificationsaccording to embodiments of the present invention.

FIGS. 13A to 13C depict aspects of pre-operative MRSE (ManifestRefraction Spherical Equivalent) according to embodiments of the presentinvention.

FIGS. 14A and 14B illustrate aspects of ablation profile modificationsaccording to embodiments of the present invention.

FIGS. 15A and 15B show aspects of pre-operative MRSE according toembodiments of the present invention.

FIGS. 16A and 16B show aspects of differences between modified targetsand original targets according to embodiments of the present invention.

FIG. 17 depicts aspects of shows post-operating secondary sphericalaberration according to embodiments of the present invention.

FIG. 18 depicts aspects of methods for generating a target shape,according to embodiments of the present invention.

FIG. 19 depicts aspects of relationships between RMS error and size of s(pixels), according to embodiments of the present invention.

FIG. 20 illustrates aspects of deconvolution methods according toembodiments of the present invention.

FIGS. 21A and 21B show aspects of ablation profile changes ormodifications according to embodiments of the present invention.

FIG. 22 illustrates aspects of induced SA according to embodiments ofthe present invention.

FIG. 23 illustrates aspects of deconvolution effects according toembodiments of the present invention.

FIG. 24 illustrates aspects of radial compensation function according toembodiments of the present invention.

FIG. 25 illustrates aspects of target shape modification according toembodiments of the present invention.

FIG. 26 shows aspects of induced SA according to embodiments of thepresent invention.

FIG. 27 illustrates aspects of low pass filter according to embodimentsof the present invention.

FIGS. 28A and 28B illustrate aspects of kernel and inverse kernelaccording to embodiments of the present invention.

FIG. 29 illustrates aspects of treatment target deconvolution accordingto embodiments of the present invention.

FIG. 30 depicts aspects of target verification according to embodimentsof the present invention.

FIGS. 31A to 31C illustrate aspects of residual error with deconvolutionaccording to embodiments of the present invention.

FIGS. 32A-32C depict aspects of expected and inversed convolved targetsaccording to embodiments of the present invention.

FIG. 33 illustrates aspects of low pass filter according to embodimentsof the present invention.

FIG. 34 illustrates aspects of post-operative SA according toembodiments of the present invention.

FIGS. 35A-35C show aspects of vision condition cases according toembodiments of the present invention.

FIG. 36 depicts post-operative SA as a function of pre-operativemanifest refraction according to embodiments of the present invention.

FIG. 37 shows aspects of kernels according to embodiments of the presentinvention.

FIG. 38 provides comparisons between observed and simulatedpost-operative outcomes according to embodiments of the presentinvention.

FIG. 39 provides comparisons between observed and simulatedpost-operative outcomes according to embodiments of the presentinvention.

FIG. 40 provides comparisons between observed and simulatedpost-operative outcomes according to embodiments of the presentinvention.

FIG. 41 depicts aspects of kernels according to embodiments of thepresent invention.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the present invention can be readily adapted for use withexisting laser systems, wavefront measurement systems, and other opticalmeasurement devices. While the systems, software, and methods of thepresent invention are described primarily in the context of a laser eyesurgery system, it should be understood the present invention may beadapted for use in alternative eye treatment procedures and systems suchas spectacle lenses, intraocular lenses, contact lenses, corneal ringimplants, collagenous corneal tissue thermal remodeling, and the like.

Turning now to the drawings, FIG. 1 illustrates a laser eye surgerysystem 10 of the present invention, including a laser 12 that produces alaser beam 14. Laser 12 is optically coupled to laser delivery optics16, which directs laser beam 14 to an eye E of patient P. A deliveryoptics support structure (not shown here for clarity) extends from aframe 18 supporting laser 12. A microscope 20 is mounted on the deliveryoptics support structure, the microscope often being used to image acornea of eye E.

Laser 12 generally comprises an excimer laser, ideally comprising anargon-fluorine laser producing pulses of laser light having a wavelengthof approximately 193 nm. Laser 12 will preferably be designed to providea feedback stabilized fluence at the patient's eye, delivered viadelivery optics 16. The present invention may also be useful withalternative sources of ultraviolet or infrared radiation, particularlythose adapted to controllably ablate the corneal tissue without causingsignificant damage to adjacent and/or underlying tissues of the eye.Such sources include, but are not limited to, solid state lasers andother devices which can generate energy in the ultraviolet wavelengthbetween about 185 and 205 nm and/or those which utilizefrequency-multiplying techniques. Hence, although an excimer laser isthe illustrative source of an ablating beam, other lasers may be used inthe present invention.

Laser system 10 will generally include a computer or programmableprocessor 22. Processor 22 may comprise (or interface with) aconventional PC system including the standard user interface devicessuch as a keyboard, a display monitor, and the like. Processor 22 willtypically include an input device such as a magnetic or optical diskdrive, an internet connection, or the like. Such input devices willoften be used to download a computer executable code from a tangiblestorage media 29 embodying any of the methods of the present invention.Tangible storage media 29 may take the form of a floppy disk, an opticaldisk, a data tape, a volatile or non-volatile memory, RAM, or the like,and the processor 22 will include the memory boards and other standardcomponents of modern computer systems for storing and executing thiscode. Tangible storage media 29 may optionally embody wavefront sensordata, wavefront gradients, a wavefront elevation map, a treatment map, acorneal elevation map, and/or an ablation table. While tangible storagemedia 29 will often be used directly in cooperation with a input deviceof processor 22, the storage media may also be remotely operativelycoupled with processor by means of network connections such as theinternet, and by wireless methods such as infrared, Bluetooth, or thelike.

Laser 12 and delivery optics 16 will generally direct laser beam 14 tothe eye of patient P under the direction of a computer 22. Computer 22will often selectively adjust laser beam 14 to expose portions of thecornea to the pulses of laser energy so as to effect a predeterminedsculpting of the cornea and alter the refractive characteristics of theeye. In many embodiments, both laser beam 14 and the laser deliveryoptical system 16 will be under computer control of processor 22 toeffect the desired laser sculpting process, with the processor effecting(and optionally modifying) the pattern of laser pulses. The pattern ofpulses may by summarized in machine readable data of tangible storagemedia 29 in the form of a treatment table, and the treatment table maybe adjusted according to feedback input into processor 22 from anautomated image analysis system in response to feedback data providedfrom an ablation monitoring system feedback system. Optionally, thefeedback may be manually entered into the processor by a systemoperator. Such feedback might be provided by integrating the wavefrontmeasurement system described below with the laser treatment system 10,and processor 22 may continue and/or terminate a sculpting treatment inresponse to the feedback, and may optionally also modify the plannedsculpting based at least in part on the feedback. Measurement systemsare further described in U.S. Pat. No. 6,315,413, the full disclosure ofwhich is incorporated herein by reference.

Laser beam 14 may be adjusted to produce the desired sculpting using avariety of alternative mechanisms. The laser beam 14 may be selectivelylimited using one or more variable apertures. An exemplary variableaperture system having a variable iris and a variable width slit isdescribed in U.S. Pat. No. 5,713,892, the full disclosure of which isincorporated herein by reference. The laser beam may also be tailored byvarying the size and offset of the laser spot from an axis of the eye,as described in U.S. Pat. Nos. 5,683,379, 6,203,539, and 6,331,177, thefull disclosures of which are incorporated herein by reference.

Still further alternatives are possible, including scanning of the laserbeam over the surface of the eye and controlling the number of pulsesand/or dwell time at each location, as described, for example, by U.S.Pat. No. 4,665,913, the full disclosure of which is incorporated hereinby reference; using masks in the optical path of laser beam 14 whichablate to vary the profile of the beam incident on the cornea, asdescribed in U.S. Pat. No. 5,807,379, the full disclosure of which isincorporated herein by reference; hybrid profile-scanning systems inwhich a variable size beam (typically controlled by a variable widthslit and/or variable diameter iris diaphragm) is scanned across thecornea; or the like. The computer programs and control methodology forthese laser pattern tailoring techniques are well described in thepatent literature.

Additional components and subsystems may be included with laser system10, as should be understood by those of skill in the art. For example,spatial and/or temporal integrators may be included to control thedistribution of energy within the laser beam, as described in U.S. Pat.No. 5,646,791, the full disclosure of which is incorporated herein byreference. Ablation effluent evacuators/filters, aspirators, and otherancillary components of the laser surgery system are known in the art.Further details of suitable systems for performing a laser ablationprocedure can be found in commonly assigned U.S. Pat. Nos. 4,665,913,4,669,466, 4,732,148, 4,770,172, 4,773,414, 5,207,668, 5,108,388,5,219,343, 5,646,791 and 5,163,934, the complete disclosures of whichare incorporated herein by reference. Suitable systems also includecommercially available refractive laser systems such as thosemanufactured and/or sold by Alcon, Bausch & Lomb, Nidek, WaveLight,LaserSight, Schwind, Zeiss-Meditec, and the like. Basis data can befurther characterized for particular lasers or operating conditions, bytaking into account localized environmental variables such astemperature, humidity, airflow, and aspiration.

FIG. 2 is a simplified block diagram of an exemplary computer system 22that may be used by the laser surgical system 10 of the presentinvention. Computer system 22 typically includes at least one processor52 which may communicate with a number of peripheral devices via a bussubsystem 54. These peripheral devices may include a storage subsystem56, comprising a memory subsystem 58 and a file storage subsystem 60,user interface input devices 62, user interface output devices 64, and anetwork interface subsystem 66. Network interface subsystem 66 providesan interface to outside networks 68 and/or other devices, such as thewavefront measurement system 30.

User interface input devices 62 may include a keyboard, pointing devicessuch as a mouse, trackball, touch pad, or graphics tablet, a scanner,foot pedals, a joystick, a touchscreen incorporated into the display,audio input devices such as voice recognition systems, microphones, andother types of input devices. User input devices 62 will often be usedto download a computer executable code from a tangible storage media 29embodying any of the methods of the present invention. In general, useof the term “input device” is intended to include a variety ofconventional and proprietary devices and ways to input information intocomputer system 22.

User interface output devices 64 may include a display subsystem, aprinter, a fax machine, or non-visual displays such as audio outputdevices. The display subsystem may be a cathode ray tube (CRT), aflat-panel device such as a liquid crystal display (LCD), a projectiondevice, or the like. The display subsystem may also provide a non-visualdisplay such as via audio output devices. In general, use of the term“output device” is intended to include a variety of conventional andproprietary devices and ways to output information from computer system22 to a user.

Storage subsystem 56 can store the basic programming and data constructsthat provide the functionality of the various embodiments of the presentinvention. For example, a database and modules implementing thefunctionality of the methods of the present invention, as describedherein, may be stored in storage subsystem 56. These software modulesare generally executed by processor 52. In a distributed environment,the software modules may be stored on a plurality of computer systemsand executed by processors of the plurality of computer systems. Storagesubsystem 56 typically comprises memory subsystem 58 and file storagesubsystem 60.

Memory subsystem 58 typically includes a number of memories including amain random access memory (RAM) 70 for storage of instructions and dataduring program execution and a read only memory (ROM) 72 in which fixedinstructions are stored. File storage subsystem 60 provides persistent(non-volatile) storage for program and data files, and may includetangible storage media 29 (FIG. 1) which may optionally embody wavefrontsensor data, wavefront gradients, a wavefront elevation map, a treatmentmap, and/or an ablation table. File storage subsystem 60 may include ahard disk drive, a floppy disk drive along with associated removablemedia, a Compact Digital Read Only Memory (CD-ROM) drive, an opticaldrive, DVD, CD-R, CD-RW, solid-state removable memory, and/or otherremovable media cartridges or disks. One or more of the drives may belocated at remote locations on other connected computers at other sitescoupled to computer system 22. The modules implementing thefunctionality of the present invention may be stored by file storagesubsystem 60.

Bus subsystem 54 provides a mechanism for letting the various componentsand subsystems of computer system 22 communicate with each other asintended. The various subsystems and components of computer system 22need not be at the same physical location but may be distributed atvarious locations within a distributed network. Although bus subsystem54 is shown schematically as a single bus, alternate embodiments of thebus subsystem may utilize multiple busses.

Computer system 22 itself can be of varying types including a personalcomputer, a portable computer, a workstation, a computer terminal, anetwork computer, a control system in a wavefront measurement system orlaser surgical system, a mainframe, or any other data processing system.Due to the ever-changing nature of computers and networks, thedescription of computer system 22 depicted in FIG. 2 is intended only asa specific example for purposes of illustrating one embodiment of thepresent invention. Many other configurations of computer system 22 arepossible having more or less components than the computer systemdepicted in FIG. 2.

Referring now to FIG. 3, one embodiment of a wavefront measurementsystem 30 is schematically illustrated in simplified form. In verygeneral terms, wavefront measurement system 30 is configured to senselocal slopes of a gradient map exiting the patient's eye. Devices basedon the Hartmann-Shack principle generally include a lenslet array tosample the gradient map uniformly over an aperture, which is typicallythe exit pupil of the eye. Thereafter, the local slopes of the gradientmap are analyzed so as to reconstruct the wavefront surface or map.

More specifically, one wavefront measurement system 30 includes an imagesource 32, such as a laser, which projects a source image throughoptical tissues 34 of eye E so as to form an image 44 upon a surface ofretina R. The image from retina R is transmitted by the optical systemof the eye (e.g., optical tissues 34) and imaged onto a wavefront sensor36 by system optics 37. The wavefront sensor 36 communicates signals toa computer system 22′ for measurement of the optical errors in theoptical tissues 34 and/or determination of an optical tissue ablationtreatment program. Computer 22′ may include the same or similar hardwareas the computer system 22 illustrated in FIGS. 1 and 2. Computer system22′ may be in communication with computer system 22 that directs thelaser surgery system 10, or some or all of the components of computersystem 22, 22′ of the wavefront measurement system 30 and laser surgerysystem 10 may be combined or separate. If desired, data from wavefrontsensor 36 may be transmitted to a laser computer system 22 via tangiblemedia 29, via an I/O port, via an networking connection 66 such as anintranet or the Internet, or the like.

Wavefront sensor 36 generally comprises a lenslet array 38 and an imagesensor 40. As the image from retina R is transmitted through opticaltissues 34 and imaged onto a surface of image sensor 40 and an image ofthe eye pupil P is similarly imaged onto a surface of lenslet array 38,the lenslet array separates the transmitted image into an array ofbeamlets 42, and (in combination with other optical components of thesystem) images the separated beamlets on the surface of sensor 40.Sensor 40 typically comprises a charged couple device or “CCD,” andsenses the characteristics of these individual beamlets, which can beused to determine the characteristics of an associated region of opticaltissues 34. In particular, where image 44 comprises a point or smallspot of light, a location of the transmitted spot as imaged by a beamletcan directly indicate a local gradient of the associated region ofoptical tissue.

Eye E generally defines an anterior orientation ANT and a posteriororientation POS. Image source 32 generally projects an image in aposterior orientation through optical tissues 34 onto retina R asindicated in FIG. 3. Optical tissues 34 again transmit image 44 from theretina anteriorly toward wavefront sensor 36. Image 44 actually formedon retina R may be distorted by any imperfections in the eye's opticalsystem when the image source is originally transmitted by opticaltissues 34. Optionally, image source projection optics 46 may beconfigured or adapted to decrease any distortion of image 44.

In some embodiments, image source optics 46 may decrease lower orderoptical errors by compensating for spherical and/or cylindrical errorsof optical tissues 34. Higher order optical errors of the opticaltissues may also be compensated through the use of an adaptive opticelement, such as a deformable mirror (described below). Use of an imagesource 32 selected to define a point or small spot at image 44 uponretina R may facilitate the analysis of the data provided by wavefrontsensor 36. Distortion of image 44 may be limited by transmitting asource image through a central region 48 of optical tissues 34 which issmaller than a pupil 50, as the central portion of the pupil may be lessprone to optical errors than the peripheral portion. Regardless of theparticular image source structure, it will be generally be beneficial tohave a well-defined and accurately formed image 44 on retina R.

In one embodiment, the wavefront data may be stored in a computerreadable medium 29 or a memory of the wavefront sensor system 30 in twoseparate arrays containing the x and y wavefront gradient valuesobtained from image spot analysis of the Hartmann-Shack sensor images,plus the x and y pupil center offsets from the nominal center of theHartmann-Shack lenslet array, as measured by the pupil camera 51 (FIG.3) image. Such information contains all the available information on thewavefront error of the eye and is sufficient to reconstruct thewavefront or any portion of it. In such embodiments, there is no need toreprocess the Hartmann-Shack image more than once, and the data spacerequired to store the gradient array is not large. For example, toaccommodate an image of a pupil with an 8 mm diameter, an array of a20×20 size (i.e., 400 elements) is often sufficient. As can beappreciated, in other embodiments, the wavefront data may be stored in amemory of the wavefront sensor system in a single array or multiplearrays.

While the methods of the present invention will generally be describedwith reference to sensing of an image 44, it should be understood that aseries of wavefront sensor data readings may be taken. For example, atime series of wavefront data readings may help to provide a moreaccurate overall determination of the ocular tissue aberrations. As theocular tissues can vary in shape over a brief period of time, aplurality of temporally separated wavefront sensor measurements canavoid relying on a single snapshot of the optical characteristics as thebasis for a refractive correcting procedure. Still further alternativesare also available, including taking wavefront sensor data of the eyewith the eye in differing configurations, positions, and/ororientations. For example, a patient will often help maintain alignmentof the eye with wavefront measurement system 30 by focusing on afixation target, as described in U.S. Pat. No. 6,004,313, the fulldisclosure of which is incorporated herein by reference. By varying aposition of the fixation target as described in that reference, opticalcharacteristics of the eye may be determined while the eye accommodatesor adapts to image a field of view at a varying distance and/or angles.

The location of the optical axis of the eye may be verified by referenceto the data provided from a pupil camera 52. In the exemplaryembodiment, a pupil camera 52 images pupil 50 so as to determine aposition of the pupil for registration of the wavefront sensor datarelative to the optical tissues.

An alternative embodiment of a wavefront measurement system isillustrated in FIG. 3A. The major components of the system of FIG. 3Aare similar to those of FIG. 3. Additionally, FIG. 3A includes anadaptive optical element 53 in the form of a deformable mirror. Thesource image is reflected from deformable mirror 98 during transmissionto retina R, and the deformable mirror is also along the optical pathused to form the transmitted image between retina R and imaging sensor40. Deformable mirror 98 can be controllably deformed by computer system22 to limit distortion of the image formed on the retina or ofsubsequent images formed of the images formed on the retina, and mayenhance the accuracy of the resultant wavefront data. The structure anduse of the system of FIG. 3A are more fully described in U.S. Pat. No.6,095,651, the full disclosure of which is incorporated herein byreference.

The components of an embodiment of a wavefront measurement system formeasuring the eye and ablations may comprise elements of a WaveScan®System. One embodiment includes a WaveScan® System with a deformablemirror as described above. An alternate embodiment of a wavefrontmeasuring system is described in U.S. Pat. No. 6,271,915, the fulldisclosure of which is incorporated herein by reference. It isappreciated that any wavefront aberrometer could be employed for usewith the present invention.

Post-Operative Aberrations

Refractive procedures may, in some cases, induce certain aberrations inan eye of a patient. For example, it is believed that laser-assisted insitu keratomileusis (LASIK) surgeries can induce high order aberrations,and in particular spherical aberration (SA). Spherical aberration is aspecial type of high order aberration that can affect night vision, andinvolves off-axis rays entering the eye with different heights of focusat different locations.

Embodiments of the present invention encompass systems and methods forreducing, eliminating, or otherwise compensating for such post-operativeinductions. For example, whereas an original target shape applied to theeye may lead to induced aberrations, it is possible to deconvolve theoriginal target shape so as to obtain a modified target shape, such thatwhen the modified target shape is applied to the eye, there are fewer orless pronounced induced aberrations.

FIG. 4 depicts aspects of a method 400 for determining a visiontreatment for an eye of a patient As shown here, the method includesreceiving (e.g. at an input) an original target profile for the eye ofthe patient as indicated by step 410. Method 400 also includes obtaininga spatial domain kernel filter as indicated by step 420. The spatialdomain kernel filter can be based on an inverse Fourier transform of aFourier domain noise filter. Further, the method may include convolvingthe original target profile with the spatial domain kernel filter asindicated by step 430. As illustrated here, method 400 also may includedetermining the vision treatment based on the convolved profile asindicated by step 440. According to some embodiments, methods mayinclude administering the vision treatment to the patient as indicatedby step 450.

FIG. 5 depicts aspects of a method for modifying a target shapeaccording to embodiments of the present invention. As shown here, amodification method 500 includes obtaining a target shape as indicatedby step 510. Often, the target shape or profile will have an opticalzone and a transition zone. In some cases, a target shape may refer toan intended optical surface designed to achieve a given refractivecorrection. A method 500 for modifying or deconvolving a target shapemay also include offsetting an inner boundary of the transition zone(e.g. by about 0.1 mm in diameter), as indicated by step 520. Further,the method may include inputting, receiving, or reading in an inversesmoothing kernel as described elsewhere herein. As illustrated by step530, methods may include applying a deconvolution to a target profile,for example as a low pass filter multiplied with the target profile asdiscussed below with reference to Equation 14. Methods may also includezeroing out an ablation profile at distances greater than the transitionzone radius, as indicated by step 540. In some cases, methods mayinclude rescaling a deconvolved target, for example as indicated by step550, so that its Zernike defocus term within the 4 mm diameter is thesame as for the original target. In some instances, the rescaling factorcan be 1.0. Optionally, methods may include elevating the entireablation profile, as depicted by step 560, so that the lowest point onthe ablation profile is zero. This elevation technique can help toensure that the ablation profile does not have negative heights. In someinstances, methods may include applying a damping multiplier (e.g.Equation 17) to the periphery of the transition zone, as indicated bystep 570. Optionally, a modification or deconvolution method can beimplemented before application of a cosine compensation step.

Post-Operative Epithelial Smoothing and Spherical Aberration

As noted above, cornea remodeling following treatment with a refractivetarget shape can induce SA, for example due to smoothing of epitheliumat the anterior surface of the eye. To develop techniques thatcompensate for such remodeling, it is helpful to simulate thepost-operative epithelium smoothing process with a model. An exemplarymodel may define the shape of the post-operative cornea surface as aconvolution of an ablation target profile with a low-pass filter (LPF),as follows:h _(post-op) =h _(pre-op) −K

T  Equation 1where T is the ablation target profile. K=K(x,y) is the LPF kernel,which has the following Fourier transform:

$\begin{matrix}{{K( {k_{x},k_{y}} )} = \frac{1}{1 + {\sigma^{2}( {k_{x}^{2} + k_{y}^{2}} )}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

K(x,y), the LPF kernel, can be considered as a spatial domainrepresentation. The Fourier transform of K(x,y) (i.e. K(k_(x), k_(y)) orF[K]), can be considered as a frequency or Fourier domainrepresentation.

According to some embodiments, the Fourier transform F[K], or K(k_(x),k_(y)), may be a squared Butterworth low-pass filter of the first order,which can be applied to the treatment target Tin order to obtain thewavefront change due to corneal smoothing. In some instances, theFourier transform of the LPF kernel can be defined by or based on asingle diffusion coefficient σ, which has a unit of length.

In some instances, the post-operative induced spherical aberration canbe computed with a Zernike decomposition of the simulated post-operativecornea surface after the smoothing, as follows:SA_(post-op)=SA_(pre-op)−SA(K

T)  Equation 3

The spherical aberration computed by Zernike decomposition of a giventarget can be represented by the function SA(T), where SA(T) refers toSA from the target T.

According to an exemplary experimental embodiment, a target for each eyein a US IDE clinical study was computed as follows:T=scale·T _(controller)  Equation 4

According to some embodiments, T_(controller) may refer to a targetcreated by production code. Such a target can be created according tovarious options. For example, the target shape can be generated based oninput such as measured pre-operative Zernike coefficients with addedflap-induced spherical aberration (e.g. flapSA). The target shape canalso be generated with or without applying a cosine correction (e.g.warping adjustment). In some cases, the target can be generated based onscaling and/or physician adjustments. Target shapes may also begenerated based on keratometry parameters. For example, if available,keratometry parameters k1, k2, k2a may be used. Optionally, for exampleif keratometry parameters are not available, default values of k1=43.5,k2=43.5, k2a=0 may be used.

It is possible to simulate the cornea thickness after smoothing using anLPF model. For example, FIG. 7 shows simulated epithelium thicknessprofiles after smoothing (High Myopia study, case ID=21011 OD, −7.4D/−1.5 D×179). For this illustration, pre-operative epithelium wasassumed uniform and 50 um thick. Corneal smoothing after a myopicablation may lead to epithelium diffusion, from high curvature areas onthe peripheral transition zone, toward the center where the curvature issmaller. As a result, the epithelium may become thicker in the centerand thinner on the periphery of the ablation target. This effect mayhelp explain partial regression after myopia refractive surgery.

Using available clinical data, a smoothed target was compared with theobserved 6M corneal change within 6 mm and 5.5 mm diameter optical zone.A diffusion coefficient σ was estimated based on the comparison. In somecases, the comparison can be performed with a linear least-square fit ofthe model to the observed SA change, as described elsewhere herein.According to some embodiments, the fitting procedure yielded anestimation of σ and its confidence interval for each value of flapSA.

Various independent estimations of σ were used, including (a) RMS matchfor low and high Myopia (6M), (b) and Hyperopia (6M-9M), and (c)slope-based estimation for low Myopia (6M). For example, FIGS. 8A and 8Bdepict optimized sigma vs. flap induced SA (simulations for clinicalstudies) for WFD=6 mm and WFD=5.5 mm, respectively. The dashed linesrepresent confidence intervals. WFD refers to a wavefront diameter.

As flap-induced aberrations typically do not depend on the type of thesubsequent treatment, it is possible to assume that the optimal valuesfor flapSA and σ can be chosen within the crossing of confidenceintervals for these three estimates (e.g. circled data points in FIGS.8A and 8B). These points can define optimal values approximately σ=0.3mm, flapSA=0.09 um for 6 mm wavefront and σ=0.45 mm, flapSA=0.05 um for5.5 mm wavefront. Some clinical observations for a flap incision withouta subsequent ablation show close values for the flap induced SA (e.g.flapSA≅=0.07 um).

It is possible to compare simulated and observed post-operative SA (e.g.with WFD=6 mm). For example, as depicted in FIGS. 9A, B, and C, anestimated diffusion coefficient σ=0.3 mm for 6 mm wavefront diameter maybe validated by comparison of simulated post-operative SA with theactual observed values. A flapSA=0.09 um was assumed for all data sets.In some embodiments, this value might be different for mechanicalmicrokeratome and IntraLase® femtosecond laser treatments. Asillustrated here, trend lines for simulated and observed data can bealmost identical for myopia and high myopia data and rather close forother data sets.

Hence, it is understood that epithelial smoothing subsequent torefractive surgery can induce SA, and that simulation of smoothing canbe helpful in developing approaches that compensate for the smoothing.In some cases, it is possible to define the shape of the post-operativecornea surface as a convolution of an ablation target profile with alow-pass filter (LPF).

In some cases, the post-operative epithelium smoothing process can besimulated by defining the shape of the post-operative cornea surface asa convolution of the ablation target profile with a low-pass filter(LPF) as follows (spatial domain):h _(post-op) =h _(pre-op) −K(x,y)

T(x,y)  Equation 5where h stands for the elevation maps,

denotes a convolution, T(x, y) is the ablation target profile and K(x,y) is a low pass filter (LPF) kernel, which has the following Fouriertransform:

$\begin{matrix}{{K( {k_{x},k_{y}} )} = \frac{1}{1 + \frac{\sigma^{2}( {k_{x}^{2} + k_{y}^{2}} )}{( {0.5\mspace{14mu}{dL}} )^{2}}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

Equation 6, which is in the Fourier domain, represents a squaredButterworth low-pass filter of the first order, which can be applied tothe treatment target in order to obtain the wavefront change due to thecorneal smoothing. It can be defined by a single diffusion coefficientσ, which has a unit of length. For some discrete case embodiments, the101×101 mesh size can be dL=0.1 mm. Based on optimizations using datafrom certain clinical trials, a sigma of 0.35 mm was determined to bestexplain that observed data.

According to some embodiments, K(x, y) is in the spatial domain, and isa Fourier transform of K(k_(x), k_(y)). Here, k_(x) and k_(y) areFourier domain or frequency domain variables. According to someembodiments, K(x, y) is an LPF kernel that can be exemplified by a101×101 matrix or by a 3-D surface expressed in matrix form where x andy are spatial domain variables.

Matching Simulation Results Vs. Observed Data

According to some embodiments, it is possible to match or comparesimulated post-operative SA with observed 6M post-operative SA usinglinear least-square fit of the model to the observed SA change byminimizing the following function:

$\begin{matrix}{F = {\sum\limits_{{all}\_{eyes}}\frac{\lbrack {{flapSA} + {{SA}( {K \otimes T} )} - ( {{SA}_{{post} - {op}} - {SA}_{{pre} - {op}}} )} \rbrack^{2}}{N}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

Here SA_(pre-op) and SA_(post-op) are spherical aberration values forpre-operational and 6M post-operative wavefront measurements, flapSA isthe immediate flap-induced SA value before the smoothing, and N is thenumber of eyes. It is possible to compute this function (F) fordifferent flapSA and diffusion coefficients, σ, and for each flapSA tofind the value σ min where fitting residual is minimal. SA (K

T) refers to the SA of the target T after LPF.

The confidence interval for the optimized σ can be roughly estimated as:

$\begin{matrix}{{\Delta\sigma} = {\frac{{std}( \lbrack {{{SA}( {K \otimes T} )} - ( {{SA}_{{post} - {op}} - {SA}_{{pre} - {op}}} )} \rbrack^{2} )}{\sqrt{N}} \cdot \frac{\mathbb{d}\sigma}{\mathbb{d}{SA}}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

Here std is a standard deviation, computed for the ensemble of eyes withthe optimized value σ=σ_(min).

Both optimized σ and its confidence interval can depend on the value offlapSA. This dependence can be computed separately for myopic (6M) andhyperopic (6M-9M) eyes, for example as depicted in FIGS. 8A and 8B.Hence, it is possible to have two independent estimations for optimizedflapSA and σ.

An alternative estimation of these values can be obtained from matchingthe simulated vs. observed trend slopes, as follows:

$\begin{matrix}{\langle \frac{{\mathbb{d}\Delta}\;{SA}^{({sim})}}{\mathbb{d}{SE}_{{pre} - {op}}} \rangle_{{all}\_{eyes}} = \langle \frac{\Delta\;{SA}^{(\exp)}}{\mathbb{d}{SE}_{{pre} - {op}}} \rangle_{{all}\_{eyes}}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

Here ΔSA=SA(K

T)−(SA_(post-op)−SA_(pre-op)). The optimized σ can provide a simulatedslope that is the same as the observed slope. A confidence interval forthis estimate can be defined as 95% confidence interval for the slope oflinear regression, as follows:

$\begin{matrix}{{\Delta\sigma} = {\frac{\mathbb{d}\sigma}{\mathbb{d}{SA}} \cdot \frac{t_{0.025} \cdot s}{s_{x}\sqrt{N - 1}}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

Here

${t_{0.025} = 1.96},{s^{2} = {\frac{N - 1}{N - 2} \cdot ( {s_{y}^{2} - {s_{x}^{2}\frac{\mathbb{d}{SA}_{{post} - {op}}}{\mathbb{d}{SE}_{{pre} - {op}}}}} )}},{s_{x} = {{stdev}( {SE}_{{pre} - {op}} )}},{s_{y} = {{{stdev}( {SA}_{{post} - {op}} )}.}}$The slope-based estimation was calculated for a Myopia study.

Offset Transition Zone

In some instances, a target shape or ablation target profile willinclude an optical zone and a transition zone. The aggregate of theoptical zone and transition zone may be referred to as an ablation zone,corresponding to the entire corneal region covered by a laser ablation.The optical zone may refer to a corneal region which received a fullintended refractive treatment. A transition zone may refer to a cornealregion outside of the optical zone but inside of the ablation zone.Often, a transition zone receives a treatment that is not strictlyoptically correct. With returning reference to FIG. 5, exemplary methodsmay also include offsetting an inner boundary of the transition zone, asindicated by step 520. According to some embodiments, an original targetshape may include a transition zone starting at about 0.25 mm inside theboundary of the optical zone. It is possible that such a target mayinduce some post-operative SA, independent of any effect corneasmoothing may have on post-operative SA. Hence, a total induced SA mayinclude a target-induced SA combined with a subsequent smoothing-inducedSA.

For example, FIG. 6A depicts post-operative values, in microns,simulated with σ=0.3 mm for study data (n=340), for SA as indicated inTable 1.

TABLE 1 Symbol Source of induced SA □ Original target shape, no cornealsmoothing (i.e. immediately after ablation) Δ Original target shape, andcorneal smoothing ⋄ Modified target shape (transition zone extended by0.1 mm), no corneal smoothing

As shown here, a target-induced SA (□) may be reduced or even completelyeliminated with a small offset of the transition zone (⋄). In somecases, the offset of the transition zone may cause sharper gradients inthe peripheral target. A 0.05 mm radial shift of the inner boundary ofthe transition zone away from the center of the optical zone, forexample as shown in FIG. 6B, (corresponding to a diameter change of 0.1mm), can make the trend slope for target-induced SA vs. pre-operative SEabout twice as small and bring the magnitude of target-induced SA (⋄)below 0.1 um level, which may be considered negligible. In someinstances, by offsetting the inner boundary of the transition zone (e.g.by about 0.1 mm in diameter), the target induced SA can be reduced byabout 50% (e.g. 0.1 mm change in diameter). As depicted here, correctingthe target induced SA can be effective to remove post-operative SA.

Deconvolution

With returning reference to FIG. 5, a method of modifying a target shapecan also include applying a deconvolution to the target profile orshape, as indicated by step 530. For example, methods may includeapplying a low pass filter (LPF) deconvolution (e.g. with σ=0.35 mm) tothe target profile. Sigma (σ) can refer to a diffusion coefficientrelated to the strength of an LPF process.

According to some embodiments, the application of a deconvolutiontransformation to an original target can operate to compensate for thearea of high curvature, which can be a significant cause ofpost-operatively induced SA.

In some instances, an LPF kernel for a deconvolution may be the same asthe one optimized to fit an observed induced post-operative SA, forexample such as those described above in connection with thepost-operative epithelial smoothing and spherical aberration. Cornealsmoothing, simulated as convolution with an identical or similar LPFkernel, can bring the cornea back to the desired shape.

In some instances, high-frequency variations may be suppressed bydiffusion or LPF convolution. Restoration of such suppressed variationsby deconvolution may introduce inaccuracies, which may also beinfluenced by a signal-to-noise level.

Embodiments of the present invention encompass the use of deconvolutiontechniques which can reduce the degree to which suppressed variationsmay introduce such inaccuracies. For example, deconvolution techniquesmay involve the use of a deconvolution filter, combining an LPF kernel,K, and a signal-to-noise ratio, SNR. The Fourier transform of such afilter can be expressed as follows:

$\begin{matrix}{{{DK}\overset{\rho}{(k)}} = \frac{K^{*}\overset{\rho}{(k)}}{{{K\overset{\rho}{(k)}}}^{2} + {SNR}^{2}}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

Here K(k) represents a Fourier transform of a LPF kernel, the asterisksrefers to a complex conjugate, and SNR is the signal-to-noise ratio.According to some embodiments, the SNR is assumed to be constant. Thevalue of SNR can define which scales will be restored by thedeconvolution, reversing diffusion effect on them. In some instances,SNR can be 0.1. If the SNR is excessively small, many small features maybe amplified. If the SNR is excessively large, only relatively largefeatures will be amplified. In exemplary embodiments of the presentinvention, SNR has a value within a range from 0 to 0.1.

If there are no noises and SNR=0, deconvolution should bring backexactly the original target, which existed before the LPF was applied.Where finite noises are present, small features may be irreversibly lostafter low-pass filtering and, therefore, deconvolution may restore theoriginal target only with a finite accuracy. The error of restorationcan be estimated with applying a LPF to a target and then usingdeconvolution to restore it and compare it with the original target.

FIG. 10A shows spherical aberration RMS errors for deconvolution fordifferent SNR values, estimated for study targets (n=340) with σ=0.3 mm,where WFD=6 mm. As depicted here, with SNR=0.1, all SA RMS errors arebelow 0.07 um level. FIG. 10B shows SA errors for a similardeconvolution, estimated for study targets (n=515) with σ=0.28 mm.

Any small and narrow dips in the measured pre-operative wavefront may beamplified by the deconvolution. This may result in small-size featuresthat are too narrow to resolve with laser pulses, which are oftenrestricted to a width of about 1 mm.

In some cases, it is not necessary or desirable to ablate these verynarrow features, as they may be flattened by the smoothing process. Whatis more, these features may also have little influence on the visionquality. In some cases, it is possible to effect the deconvolution so asto neglect or minimize these features and amplify only relativelylarge-scale features of the ablation target. For example, this can bedone by optimizing the SNR value in a deconvolution process. It has beenfound that by using SNR≧0.1, for example, any features smaller than 0.5mm are not amplified by deconvolution. Hence, SNR=0.1 may be used adefault parameter.

A deconvolved target typically has an oscillating profile at theperiphery. These oscillations may be mainly caused by boundaries betweenthe optical zone, transition zone, and an edge of the finite-sizetarget, where either the target profile or its derivatives have sharpchanges.

Embodiments of the present invention encompass the use of deconvolutionand related techniques to compensate for the post-operative induction ofhigh order aberrations (HOAs), and in particular spherical aberration(SA). Accordingly, the visual quality of patients receiving treatmentsaccording to these techniques provides desirable results, particularlyin the management of night vision symptoms. Often, deconvolutionprocedures will result in treatment target shape changes near theperiphery of the optical zone. For example, within a central 4 mm area,the refraction of a modified target shape may be similar or identical tothat of an original target shape.

According to some embodiments, to obtain a new or modified target shape,a deconvolution process can be employed as follows:

$\begin{matrix}{T_{new} = {{K_{INV} \otimes T_{current}} = {{F\lbrack \frac{K*( {k_{x},k_{y}} )}{{{K( {k_{x},k_{y}} )}}^{2} + {SNR}^{2}} \rbrack} \otimes T_{current}}}} & {{Equation}\mspace{14mu} 12}\end{matrix}$

where F(•) stands for a Fourier transform, * denotes a complexconjugate, T_(current) is an original treatment target, T_(new) is thenew target that is intended to remove the post-operative SA, and K_(INV)is the inverse kernel of K. The SNR can be used to prevent or inhibitnoise amplification and oscillation at the edge. In some instances, aSNR value of 0.1 may be suitable for practical purposes. To prevent oras a substitute for real-time calculation of the Fourier transforms, theinverse kernel K_(INV) can be pre-calculated and applied in real-time asa look-up table or a resource file. A suitable SNR value can prevent thedenominator from being zero or excessively small, which may otherwiseresults in the matrix quotient being unreasonably large.

According to some embodiments, an inverse kernel can be exemplified as aconvolution kernel that operates like a deconvolution procedure. In thissense, a deconvolution operation may be considered to be an inverseprocedure of a convolution operation.

Embodiments of the present invention encompass techniques forcalculating an inverse smoothing kernel K_(INV). Whereas a low passfilter (e.g. Butterworth kernel) such as K(x, y) is in the Fourierdomain, the inverse kernel is in the spatial domain. Instead ofimplementing a Fourier transform, it is possible to perform a spatialconvolution implemented as multiplication.

In some cases, embodiments encompass rapid convolution calculations(e.g. in the order of several milliseconds) for UI (user interface)manipulation, in a practical implementation. A normal implementation fora spatial 2-D convolution may involve four netted loops each with 101elements. Such embodiments may be related to the 101×101 mesh size casesdiscussed above in the paragraph following Equation 6. A 2-D spatialconvolution can be written as follows:

$\begin{matrix}{{T_{new}( {i,j} )} = {{T_{current} \otimes K_{INV}} = {\sum\limits_{k = {- \infty}}^{\infty}{\sum\limits_{l = {- \infty}}^{\infty}{{T_{current}( {{i - k},{j - l}} )}{K_{INV}( {k,l} )}}}}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$where K_(INV) is the 2-D inverse smoothing kernel. In some cases,K(k_(x), k_(y)) may be a Butterworth of the first kind, and its inversemay have an actual size that is only a few pixels wide. Therefore,Equation 13 may be rewritten as follows:

$\begin{matrix}{{T_{new}( {i,j} )} = {{T_{current} \otimes K_{INV}} = {\sum\limits_{k = {- s}}^{s}{\sum\limits_{l = {- s}}^{s}{{T_{current}( {{i - k},{j - l}} )}{K_{INV}( {{51 + k},{51 + l}} )}}}}}} & {{Equation}\mspace{14mu} 14}\end{matrix}$where the inverse kernel size is treated as (2s+1)×(2s+1) in size. Whens=17, or the inverse kernel frame size of 35×35, RMS error usingEquation B is about 0.01 microns. With s=37, use of Equation 14 may beabout 7 times faster than Equation 13, but the error is within 0.001microns. FIG. 19 shows the relationship between the RMS error and thesize of s (pixels), with a simulation of 515 eyes. This figure depictsthe RMS error as a function of s when Equation 14 is used (e.g. incontrast to Equation 17 as discussed below).

Zero Out

With returning reference to FIG. 5, a method of modifying a target shapecan also include zeroing out an ablation profile at distances greaterthan the transition zone radius, as indicated by step 540.

Typically, no ablation is performed beyond the end of transition zone.Hence, it is possible to zero-out the ablation profile at distancesgreater than the transition zone outer radius, R_(TZ), as discussedelsewhere herein, for example with regard to FIGS. 12C and 12D.

A zeroing-out procedure can be included, so as to prevent artifacts andthe like that might occur as a result of performing convolution ordeconvolution. For example convolution or deconvolution mayinadvertently or unintentionally introduce nonzero or negative values atpositions outside of the transition zone. A zeroing-out operation can beinstituted as a safeguard, so as to ensure that such non-zero ornegative values are removed, which could otherwise cause complicationsfor a tissue ablation protocol.

Rescaling Deconvolved Target

As shown in FIG. 5, a method of modifying a target shape can alsoinclude rescaling a deconvolved target, as indicated by step 550. Forexample, a deconvolved target can be rescaled so that its Zernikedefocus term within a 4 mm diameter is the same as that for an originaltarget. In this way, the spherical equivalent refraction of a modifiedor deconvolved target can be the same as that for an original target. Insome instances, a rescaling procedure can be performed to ensure thatthe refractive power for a deconvolved target is the same as that for anoriginal target. In some cases, the refractive power for a deconvolvedtarget is the same as that for an original target and no rescaling stepis performed.

According to some embodiments, an original target shape may performadequately for correcting or treating refraction errors, and hence amodified target shape based on the original target shape may begenerated so that the refraction of the modified target is the same asfor the original target. This can be achieved, for example, by rescalingof the deconvolved target so that its defocus Zernike term within the 4mm area (which defines wavefront-based SE) is the same as for thecurrent target. A rescaling coefficient, which is the ratio of thedefocus terms for the current and de-convolved targets, may be expressedas follows:

$\begin{matrix}{{rescale} = \frac{{SE}_{current}}{{SE}_{{de} - {conv}}}} & {{Equation}\mspace{14mu} 15}\end{matrix}$

The rescaling coefficient may be close to 1, and distributed as shown inFIGS. 11A and 11B. For example, a rescaling coefficient may have a meanvalue of 1.003, such as that which was found for US IDE studies. In suchinstances, rescaling may not be needed, in practical terms. In norescaling is performed, then resulting refraction errors may be below0.1 D, for example as shown in FIG. 11A. Hence, it may be possible toneglect or ignore such small values. FIG. 11B shows a distribution of SEre-scaling coefficients and refraction errors without rescaling for thestudies (n=340).

According to some embodiments, deconvolution may also affect thecylinder refraction. A magnitude of this effect is illustrated in FIGS.12A and 12B. Here, it is possible to see a comparison of X, Y componentsof astigmatism for an original target and a deconvolved target(simulated for the studies, n=340). The deconvolved targets showslightly higher astigmatism, as compared with the original targets,although the difference is less than 1%.

According to some embodiments, a current or original target T_(current)yields good matching to low order aberrations, and a scaling can beperformed such that the refractive spherical equivalent over 4 mm of thenew or modified target is the same as that of the current or originaltarget. Exemplary studies have shown that such a scaling factor is aboutunity. Therefore, a scaling factor of 1.0 can be assumed in some cases.

Elevating Ablation Profile

As shown in FIG. 5, a method of modifying a target shape can alsoinclude elevating an ablation profile, as indicated by step 560. Forexample, in order to make all ablation values be non-negative, it ispossible to elevate the entire ablation profile so that the lowest pointon the ablation profile is zero or otherwise non-negative. In this way,the ablation profile can be generated so that it does not have negativeheights.

Damping Periphery Of Transition Zone

As shown in FIG. 5, a method of modifying a target shape can alsoinclude damping a periphery of a transition zone, as indicated by step570. For example, a damping multiplier or multiplication factor may beapplied which suppresses the fluctuations of the periphery of the targetshape. In some embodiments, after certain adjustments are made (e.g.such as the adjustment discussed above), a peripheral part of theablation profile may have a small bump, which may be the result of acut-off at the end of the transition zone. Ablating such a bump mayrequire a sequence of many small laser pulses around the transition zoneperiphery. In some cases, this may cause a substantial reduction ofspeed in the entire ablation process. In some cases, the bump may not beneeded because it lies away from the optical zone and its influence onthe wavefront within the optical zone after smoothing may be verylimited. Embodiments of the present invention encompass the applicationof a damping multiplier to the periphery of the transition zone,starting from the distance R_(b)=R_(TZ)−0.5 mm, as follows:

$T = {T \cdot \{ \begin{matrix}\frac{R_{TZ} - R}{R_{TZ} - R_{b}} & {R > R_{b}} \\1 & {R<=R_{b}}\end{matrix} }$

Such a damping multiplier or factor can be used to eliminate or diminishthe bump.

FIG. 12C shows an X cross-section of modifications of an ablationprofile, and FIG. 12D shows a Y cross-section of modifications of anablation profile. In some embodiments, modifications of an ablationprofile (e.g. high myopia study, case ID=21011 OD) may include targetdeconvolution with σ=0.35 mm, as well as an elevation modification, or acut-off beyond the transition zone.

In some cases, a different wavefront diameter may use or benefit from adifferent diffusion coefficient (e.g. for an LPF model) to matchpost-operative measurements. In some cases, it is possible to use anapproximated value of σ=0.35 mm, which is between optimized values for 6mm and 5.5 mm wavefront diameters, as discussed elsewhere herein. Usinga diffusion coefficient such as this for the target deconvolution, it ispossible to predict or calculate a substantial reduction of induced SAfor both WFD=6 mm and WFD=5 mm and also additional ablation depthrequirement. For example, FIG. 13A depicts a simulated post-operative SAfor a 6 mm wavefront, FIG. 13B depicts a simulated post-operative SA fora 5.5 mm wavefront, and FIG. 13C depicts an extra ablation that maybenefit a deconvolved target. As such, these figures demonstrate theeffect of deconvolution on post-smoothing SA and on additional maximumablation depth.

Because deconvolution may amplify noises, the tail or outer periphery ofthe ablation profile may have some bumps. To remove such bumps, adamping multiplier can be applied as

$\begin{matrix}{T^{\prime} = {T \cdot \{ \begin{matrix}{2( {R_{TZ} - R} )} & {R > R_{b}} \\1 & {R \leq R_{b}}\end{matrix} }} & {{Equation}\mspace{14mu} 17}\end{matrix}$where T′ is the new target after damping, T is the target after Equation14 and R is a variable in radius. R_(TZ) is the transition zone radius,and the cutoff radius R_(b)=R_(TZ)−0.5 mm. This damping multiplier caneffectively and substantially eliminate the bumps.

Results and Data Analysis

Based on certain codes for treatment target creation, the following twophases of simulation studies were conducted. A first phase involvedoptimizing a one-parameter diffusion coefficient such that it bestexplains the clinically observed 6M post-operative spherical aberrationswith the same surgical parameters as these eyes were treated. A secondphase involved verifying that with the use of an optimized diffusioncoefficient, the expected post-operative spherical aberration issignificantly reduced when a deconvolution algorithm is used.

Optimization of a diffusion coefficient was based on data from variousclinical studies and trials, as well as data from commercial sites. Onlyeyes with pre-operative and 6M (3M for iDesign™ system) post-operativewavefront measurements with at least 6 mm diameter were used. As such,340 eyes were from the study, 169 eyes from the commercial sites, and 39eyes from iDesign™ system based study. Of the 340 eyes from US IDE, 158were in the low to moderate myopia cohort, 75 in the high myopia cohort,26 from hyperopia cohort, 47 from the monovision cohort (dominant eyesonly), and 34 from the mixed astigmatism cohort.

As explained elsewhere herein, a comparison between a simulated and anobserved post-operative spherical aberration can be performed for agiven diffusion coefficient. An optimization process was chosen suchthat the simulated post-operative spherical aberration has asubstantially identical slope as compared with a pre-operative sphericalequivalent to that of the observed post-operative spherical aberration.

Because of variations of the sample size in different cohorts, the 95%confidence bands are different for different cohort. A small overlaparea can be identified for these 95% confidence bands. The optimizeddiffusion coefficient of 0.35 mm was obtained from the overlap area.

According to some embodiments, deconvolution, which can be used toreduce post-operative spherical aberrations, is a physical-model-backedapproach. It is based on the smoothing effect observed from the clinicaldata. Therefore, not only can it account for the increase of thepost-operative spherical aberration, but it can also account for theinduction of other high order aberrations, such as coma, secondaryastigmatism, and secondary spherical aberration. Furthermore, asdiscussed elsewhere herein, it provides a smaller ablation depth ascompared with other techniques (e.g. larger optical zone, largerkeratometric values) used to target the same level of sphericalaberration reduction.

Many of the target shape modification discussed herein can operate tochange a peripheral area of the target so as to reduce the induction ofSA. It is possible to compare such methods, for example when theirparameters are selected to generate a small slope of SA vs. SE trend, asindicated in Table 2. The parameters in this table were selected for thesimulation to achieve a slope of SA vs. SE trend that is about the sameas the slope from the observed clinical data.

TABLE 2 Modification SA vs. SE <SA> std(SA) max |SA| <extraH> max extraHparameter trend slope um um um um um Current target −0.04 0.16 0.16 0.580.0 0.0 dOZ, mm 0.4 −0.01 −0.01 0.10 0.31 11.02 26.0 dK, D 25 −0.01−0.04 0.11 0.33 9.90 25.9 sigma, mm 0.35 0.01 −0.03 0.09 0.29 7.24 17.9

Table 2 provides a comparison of three methods of target modifications,simulated for data from the studies. Parameters for each modificationmethod were chosen to bring the magnitude of simulated slope ofpost-operative SA vs SE trend line down to 0.01. The simulated averagepost-op SA (<SA>), the worst case SA (max |SA|, the average extraablation depth (<extraH>), and the worst case (max extraH) are alsoshown. Sigma (σ) is a diffusion coefficient related to the strength ofan LPF process, described elsewhere herein. As shown in Table 2, adeconvolution method (sigma) can virtually eliminate both the mean SAand the SA vs. SE trend slope. Similarly, a widened optical zone method(dOZ) and a cosine correction adjustment method (dK) can also virtuallyeliminate both the mean SA and the SA vs. SE trend slope. Compared withwidened optical zone and cosine adjustment methods, deconvolutiontechniques often require lower amounts of ablation, and hence canprovide useful solutions where saving or maintaining more tissue isdesired.

FIG. 14A shows an X cross-section of modifications of an ablationprofile, and FIG. 14B shows a Y cross-section of modifications of anablation profile. These modifications of an ablation target aresimulated for a high myopia study (study ID=21011 OD, −7.4 D/−1.5D×179°). Simulation was performed for a wider optical zone approach(dOZ=0.4 mm), an adjusted cornea curvature for cosine correctionapproach (dK=25 D), and a deconvolution approach (σ=0.35 mm). Whenevaluating the expected post-operative SA, it may be helpful to considerthat simulations may only show the changing SA vs SE trend line afterthe target modification. In reality the post-operative SA may deviatefrom the trend line due to some other factors which are not accountedfor. These deviations can be estimated for the current target asfollows:δSA=SA_(observed) ^((6M))−SA_(simulated) ^((post-op))  Equation 18

Assuming that the same deviations from the trend line can apply to amodified target, it is possible to add δSA to the simulatedpost-operative SA values of every modified target, which can provide arealistic estimate of post-operative distribution of SA. For example,FIGS. 15A and 15B, depict post-operative SA for observed study data(n=340) and expected post-operative SA for de-convolved targets,simulated with σ=0.35 mm for the same eyes, for a 6 mm wavefront and 5.5mm wavefront, respectively.

In addition to piston differences which may be present between theoriginal and modified targets, there may be other shape differences aswell. According to some embodiments, the following metrics can be usedto compare shape differences:Δ=(H−max(H))−(H _(current)−max(H _(current)))  Equation 19where H refers to ablation depth or target height.

As illustrated in FIGS. 16A and 16B, target shapes subsequent tosmoothing for two modification methods, namely widening optical zone(dOZ) and deconvolution (sigma) are almost identical within the 6 mmoptical zone. These figures show the differences (i.e. X and Ycross-sections, respectively) between a modified target and an originaltarget, subsequent to smoothing, simulated for a high myopia case(ID=21011 OD, −7.4 D/−1.5 D×179°). Simulations were performed for awider optical zone (dOZ=0.4 mm), an adjusted corneal curvature forcosine correction (dK=25 D), and a deconvolution (δ=0.35 mm).

A cosine adjustment can make a different shape with a substantiallyhigher secondary spherical aberration, as depicted in FIG. 17. In somecases, software or systems may allow both a user-defined optical zoneand a user-defined adjustment of corneal curvature (e.g. defining thecosine correction), and these two adjustments can be used for validationfor a deconvolution technique. In some cases, a wider optical zone, mayprovide a closer approximation than a curvature adjustment. FIG. 17shows a post-operating secondary spherical aberration (WFD=6 mm),simulated for study data (n=340). Simulation was performed for originaltargets and for modified targets with a wider optical zone (dOZ=0.4 mm),an adjusted corneal curvature for cosine correction (dK=25 D), and adeconvolution (σ=0.35 mm).

In sum, the three methods for modification of an ablation target(widening optical zone, adjusting cosine correction, and deconvolution)are capable of eliminating a systematic trend in post-operativelyinduced spherical aberration. As shown here, the ablation profiles forthese modifications can present different depths, and deconvolution canprovide a technique which results in a maximum of tissue retention. Thatis, the amount of ablation associated with deconvolution is smaller thanthat of the other methods. In some instances, widened optical zone anddeconvolution techniques may yield almost identical corneal shapes aftersmoothing. In some cases, a widened optical zone technique (e.g. basedon a user-defined optical zone) may be used as a validation for adeconvolution technique.

Treatment Target Creation

As noted elsewhere herein, a treatment target shape may represent orcorrespond to an intended optical surface that is designed to achieve aparticular refractive correction. FIG. 18 depicts a method 1800 forgenerating a target shape, according to embodiments of the presentinvention. Method 1800 may include obtaining a wavefront correspondingto a pupil plane, as indicated by module 1805. For example, for targetcreation, the input can be a Fourier-based wavefront, which representsthe ocular aberrations on the pupil plane. Typically, a laser ablationis performed on the corneal surface, and hence to obtain the targetshape the ocular aberrations are propagated from the pupil plane to thecorneal surface. Accordingly, methods may include propagating thewavefront, as indicated by step 1810, and obtaining a wavefrontcorresponding to a corneal plane, as indicated by step 1815. Anyphysician adjustments or nomogram adjustments can also be represented onthe corneal surface first before they are combined with the ocularaberrations. Hence, the process of obtaining a wavefront at the cornealplane may also be based on an internal sphere adjustment, as indicatedby step 1820, or on a physician adjustment (e.g. Sph+Cyl), as indicatedby step 1825, or both.

In some instances, parameters such as optical zone size and the ablationzone size, which may be user-defined, can be used to determine theablation or target shape within such zones. Thus, the process ofobtaining a raw or original target shape, as indicated by step 1830, maybe based on a selection or definition of an optical zone, an ablationzone, or both, as indicated by step 1835.

A deconvolution technique can be used to deconvolved the raw or originalshape, so as to obtain a deconvolved shape, as indicated by step 1840.Such a deconvolution can operate to reduce post-operative sphericalaberration. Once the deconvolved shape is obtained, a scaling factor canbe applied, as indicated by step 1845, and a cosine effect modificationthat compensates for the loss of energy due to the curved cornea can beapplied, as indicated by step 1850. Hence, the final target shape can bedetermined based on the deconvolved shape, as indicated by step 1855,optionally considering a scaling factor, a cosine effect, or both.

In some instances a nomogram adjustment can be applied, as indicated bystep 1860, when obtaining the final target shape. Following creation ofthe final or modified target shape, as indicated by step 1855, thetarget shape can be transmitted to a treatment table generation engine.

Exemplary Techniques for Target Shape Deconvolution

As explained elsewhere herein, treatment target shapes can lead toinduced aberrations, and deconvolution can be applied to such treatmenttarget shapes so as to reduce or inhibit the induced aberrations.

FIG. 20 depicts aspects of a deconvolution method 2000 for a targetshape, according to embodiments of the present invention. As illustratedhere, method 2000 of deconvolving a target shape may include obtaining amesh size as indicated by step 2005 and obtaining a diffusioncoefficient as indicated by step 2010. Method 2000 may also includeobtaining a complex matrix, in Fourier domain, based on a mesh size anddiffusion coefficient as indicated by step 2015.

Complex Matrix

According to some embodiments, a complex matrix K(k_(x), k_(y)) can beapplied to a treatment target to obtain a wavefront change due tocorneal smoothing The complex matrix can be considered to represent athree dimensional matrix in a Fourier or frequency domain. In somecases, the complex matrix may be a squared Butterworth low-pass filterof the first order. Other types of low-pass filters may be suitable foruse with embodiments of the present invention. In some cases, a low-passfilter may refer to a function or operation that makes details smootherby suppressing high spatial frequency information.

In some instances, the Fourier domain complex matrix can be expressed asfollows:

$\begin{matrix}{{K( {k_{x},k_{y}} )} = \frac{1}{1 + \frac{\sigma^{2}( {k_{x}^{2} + k_{y}^{2}} )}{( {0.5{dL}} )^{2}}}} & {{Equation}\mspace{14mu} 20}\end{matrix}$where σ represents a diffusion coefficient, k_(x) and k_(y) representfrequency domain variables, and dL represents a mesh size. Optionally,the diffusion coefficient σ can have a value of 0.35 mm and the meshsize dL can have a value of 0.1 mm. In some instances, the diffusioncoefficient can have a value with a range from about 0.2 to about 0.5(see, e.g. FIG. 8A). In some instances, the diffusion coefficient canhave a value with a range from about 0.33 to about 0.4 (see, e.g. FIG.33).

In some instances, the term Fourier transform as used herein may referto a transform operation. In some instances, the term Fourier transformas used herein may refer to a complex valued function produced by atransform process.

Mesh Size

In an exemplary discrete case, a complex matrix K (k_(x), k_(y)) can bebased on a 101×101 mesh size of dL=0.1 mm. Often, such matrix formats(e.g. 101×101) are used when characterizing treatment planning. In somecases, a mesh size or dL may refer to the spacing or spatial distancebetween two neighboring pixels. In some cases, dL may refer to the pixelresolution in the kernel, which can be 101×101 in pixel frame size or 10mm×10 mm in space. When a discrete Fourier transform is involved, it ispossible to represent the frame in 101×101, although it may no longer be0.1 mm because it is in frequency domain (more like cycles per degree).Hence, dL may involve a 0.1 mm spacing in the spatial domain.

In some instances, selection of a kernel or matrix format may representa balance between accuracy and speed concerns. For example, a largerkernel or matrix format such as 101×101 may provide greater relativeaccuracy and lower relative speed, whereas a smaller kernel or matrixformat such as 25×25 may provide lower relative accuracy and greaterrelative speed.

Diffusion Coefficient

As noted above, a complex matrix can also be based on a diffusioncoefficient σ. Typically, a diffusion coefficient σ has a unit oflength. This parameter can describe the strength of corneal smoothingduring and after a refractive surgical procedure, and as such can beconsidered as a biologically related parameter. The parameter can beused to characterize a single individual, or a group of individuals.Based on the analysis of results from several clinical trials, it hasbeen discovered that a diffusion coefficient σ of 0.35 mm is consistentwith such observed data. In some instances, a diffusion coefficient canhave a value within a range from about 0.2 mm to about 0.5 mm. In someinstances, a diffusion coefficient can have a value of about 0.3 mm.

Because a Fourier domain complex matrix can be based on the mesh size,the diffusion coefficient, or both, it follows that a correspondingspatial domain kernel filter, as discussed elsewhere herein can also bebased on the mesh size, the diffusion coefficient, or both.

According to some embodiments, an LPF can be used to emulate thediffusion of corneal tissue cells. Exemplary techniques may involveestimating or receiving a diffusion coefficient value, and using thatvalue to effect a compensation for a high order aberration beforeadministering a treatment such as a laser vision corrective procedure.By pre-compensating for high order aberrations, it is possible to obtainan outcome with a reduced amount of high order aberrations.

Diffusion coefficients may be evaluated based on simulations. Forexample, a diffusion coefficient σ value can be selected for applicationto clinical data in a deconvolution procedure as described herein, andthe expected outcome (e.g. deconvolved target shape) can be compared tothe actual outcome (e.g. clinical data). The diffusion coefficient canbe adjusted or optimized so as to reduce or minimize variance or astandard deviation in the comparison results. Exemplary adjustment oroptimization techniques are described elsewhere herein, for example inconnection with FIGS. 25 to 28A.

Relatedly, embodiments encompass systems and methods for adjustingrefractive surgery parameters, which may include a diffusioncoefficient, for use in a vision treatement. An exemplary method mayinclude inputting or receiving a refractive case, determining a modeloptical surface shape based on the refractive case and a set ofrefractive surgery system parameters, comparing the refractive case andthe model optical surface shape to determine an aberration induced bythe set of refractive surgery system parameters, adjusting the set ofrefractive surgery system parameters so as to inhibit the inducedaberration, and administering the refractive treatment to a patient. Therefractive treatment can be based on the adjusted set of refractivesurgery system parameters.

Matrix Quotient

As depicted by step 2020, methods may include calculating a matrixquotient, where the dividend includes a conjugate of a Fourier domaincomplex matrix (e.g. K*(k_(x), k_(y)), and the divisor includes the sumof a squared modulus of the Fourier domain complex matrix and a signalto noise ratio value. In some cases, the signal to noise ratio value maybe a squared value. An exemplary matrix quotient can be expressed asfollows:

$\begin{matrix}{\lbrack \frac{K*( {k_{x},k_{y}} )}{{{K( {k_{x},k_{y}} )}}^{2} + {SNR}^{2}} \rbrack,} & {{Equation}\mspace{14mu} 21}\end{matrix}$

In some cases, the denominator or divisor of the matrix quotient can becharacterized at least in part by the expression |K(k_(x), k_(y))|^(n),where n is an integer having a value of 2 or more. In some cases, thedenominator or divisor of the matrix quotient can be characterized atleast in part by the expression [|K(k_(x), k_(y))|^(n)+SNR²] where n isan integer having a value of 2 or more and SNR represents a signal tonoise ratio value. Equation 21 may refer to a filtering process that isin the frequency domain. A complex conjugate may be part of thefiltering process.

Spatial Domain Kernel Filter

As depicted by step 2025, methods may also include obtaining a kernelfilter, in the spatial domain, based on an inverse Fourier transform ofthe matrix quotient. An exemplary kernel filter can be expressed asfollows:

$\begin{matrix}{F\lbrack \frac{K*( {k_{x},k_{y}} )}{{{K( {k_{x},k_{y}} )}}^{2} + {SNR}^{2}} \rbrack} & {{Equation}\mspace{14mu} 22}\end{matrix}$

In some cases, the kernel filter of Equation 22 can be provided as apre-calculated or pre-defined matrix, and can be used or saved as alookup table. As discussed elsewhere herein, this kernel filter can alsobe referred to as an inverse kernel K_(INV). Optionally, this kernelfilter can be referred to as K (x, y). This spatial domain filter orinverse kernel can also be provided as a low pass filter, such as aButterworth or Gaussian filter. Optionally, the spatial domain kernelfilter can present a grid or matrix that reflects how the filtered valueof a pixel depends on neighboring pixel values, and is independent ofthe target shape.

Convolving Raw Target

As depicted by step 2035, methods may include convolving a raw ororiginal target shape with the spatial domain kernel filter. Optionally,methods may include receiving, at an input, an original target profileor shape for the eye of the patient, as indicated by step 2030. As shownhere, the spatial domain kernel filter can be based on an inverseFourier transform of a Fourier domain noise filter, for example, whichmay be based on a conjugate of a Fourier domain complex matrix, on amodulus of a Fourier domain complex matrix, or on a combination thereof.In some instances, a Fourier domain noise filter can be characterized byfraction having a numerator comprising a conjugate of a Fourier domaincomplex matrix and a denominator comprising a modulus of the Fourierdomain complex matrix. Method 2000 indicates that an original targetshape T_(current) (x, y) can be convolved with a spatial domain kernelfilter so as to obtain a deconvolved shape T_(new) (x, y), as indicatedby step 2040. In some instances, the deconvolved shape 2040 emphasizescurvature changes, or corners, sharp edges, sharp transitions, and thelike. In some cases, methods may involve the application of a low passfilter deconvolution to a target profile having a slightly extendedoptical zone. In some instances, parameters of a low pass filter can beoptimized by comparing an LPF model prediction against observed clinicaldata.

Other Refinements

As depicted by step 2045, methods may include additional refinements ofa shape prior to transmitting the shape to a treatment table engine. Forexample, a convolved profile may include a transition zone radius, andexemplary techniques may include zeroing the convolved profile atlocations outside of the transition zone radius. In some cases, anoriginal target profile may have an original refractive sphericalequivalent value within a 4 mm diameter area, and the convolved targetprofile may have a target refractive spherical equivalent value within a4 mm diameter area, and method 2000 may include scaling the originalrefractive spherical equivalent with the target refractive sphericalequivalent value. In some cases, methods may include elevating theconvolved profile so that a lowest point on the convolved profile iszero or greater. In some cases, a convolved profile may include atransition zone radius, and methods may involve applying a dampingmultiplier at or near the transition zone radius. In some instances,refinement can be performed prior to, or subsequent to, deconvolution,with an equivalent effect.

As discussed elsewhere herein, a deconvolved target may have anoscillating profile at the periphery. Such oscillations may be caused byboundaries between the optical zone, transition zone, and edge of thefinite-size target, where either the target profile or its derivativeshave sharp changes. In some instances, it may be helpful to elevate theentire ablation profile so that the lowest point on the ablation profileis zero, or so that all ablation values are non-negative. What is more,it may be helpful to zero-out the ablation profile at distances greaterthan the transition zone radius, R_(TZ), where no ablation is desiredbeyond the end of the transition zone. Such refinements are illustratedin the X and Y target cross-sections of FIG. 21A, which depictsmodifications of an ablation profile (high myopia study, case ID=21011OD) including deconvolution (σ=0.28 mm), elevation, and cut-off beyondthe transition zone. In some cases, after such refinements oradjustments are made, only the peripheral curvature will be changed, forexample as depicted in FIG. 21B, which shows a change of ablationprofile after target deconvolution (High Myopia study, case ID=21011,OD, −7.4 D/−1.5 D×179 deg).

In some instances, an original target shape may operate to effectivelyaddress refraction errors, and hence it may be desirable to maintain therefraction of the modified target at the same value as the refraction ofthe original target. This can be done with rescaling of the deconvolvedtarget so that its defocusing term within the 4 mm area is the same asfor the original target.

In addition to, or following some or all of the above mentionedadjustments, the peripheral part of the ablation profile may have asmall bump, which results mainly from the cut-off at the end of thetransition zone, for example as depicted in FIG. 21A. Ablating such abump may involve application of a sequence of many small laser pulsesaround the transition zone periphery. In some instances, this may leadto a substantial slow-down of the entire ablation process. Yet this bumpmay be unnecessary, because it lies away from the optical zone and itsinfluence on the wavefront within the optical zone shall be rather smallafter healing. With this consideration, it is possible to apply adamping multiplier to the periphery of the transition zone, as describedelsewhere herein.

Spherical Aberration and Related Topics

As discussed elsewhere herein, spherical aberration (SA) may be inducedby a target shape, a healing effect, or a combination thereof. In somecases, it is possible to reduce or even completely eliminatetarget-induced SA by implementing a small offset of the transition zone.In some original target shapes, the inner boundary of the transitionzone is located within the optical zone, e.g. at about 0.25 mm from theedge of the optical zone. In addressing target-induced SA, it may behelpful to shift the transition zone boundary, by moving it farther fromthe center of the optical zone. In this way, the target-induced SA canbe decreased, although squeeze the transition zone and cause sharpergradients in the peripheral target. In some instances, this may meanthere will be a narrower transition zone band. In some instances,shifting the thegg inner boundary of the transition zone away from thecenter of the optical zone by a distance of about 0.1 mm can operate toreduce the target-induced SA to a level below 0.1 um, which may beconsidered negligible.

FIG. 22 shows a simulated induced SA immediately after ablation(target-induced) and after healing (total) for a target with an innerboundary of the transition zone shifted outward by 0.1 mm, using ahealing model where σ=0.28 mm. As shown here, after healing, the totalSA reached a level of about 0.3 um.

In order to compensate for the spread of the high curvature, which is amain cause of post-healing induced SA, it is helpful to apply adeconvolution transformation to the original target. In some cases, theLPF core for deconvolution is the same as the one optimized to fitobserved induced post-operative SA. Then healing, simulated asconvolution with the same LPF core, can bring the healed cornea back tothe desired shape.

FIG. 23 shows the effect of deconvolution on post-healing SA (leftpanel) and additional maximum ablation depth (right panel) simulatedwith σ=0.28 mm for studies (n=515). Relatedly, Table 3 shows simulatedchanges in post-healing SA and extra ablation, caused by deconvolutionand additional adjustments of an original target. Statistics forstudies: Myopia and High Myopia (n=327), Hyperopia (n=43), and allstudies together (n=515).

TABLE 3 old new SA(SE) SA(SE) max <extra max Slope Slope <SA> |SA| Abl>extAb Myopia & HM −0.04 −0.01 0.01 0.08 4.3 8.9 Hyperopia −0.09 −0.02−0.05 0.11 3.5 7.6 All US IDE −0.04 −0.01 0.00 0.11 4.2 9.9

FIG. 24 depicts a radial compensation function (RCF) for a deconvolvedtarget in a high myopia case, according to embodiments of the presentinvention. Specifically, a radial compensation function was calculatedfor a deconvolved target corresponding to a High Myopia study (caseID=21011 OD, −7.4 D/−1.5 D×179 deg.). As shown here, the RCF is almostflat in the central part and decreases in the periphery.

FIG. 25 schematically illustrates techniques for obtaining andimplementing a modified target shape, according to embodiments of thepresent invention. As shown here, study data can be used to deriveparameters of a kernel for simulating a low-pass filtering process, forcorneal healing and the like. Embodiments may also include optimizingthe parameters by using a clinical data set. These techniques may alsoinvolve evaluating the extent to which observed spherical aberration isattributed to error, due to an imperfect optical treatment shape. Insome instances, methods may also include addressing target shape inducedSA by providing transition zone adjustments, optical zone extensionadjustments, or both. In some cases, a deconvolution (e.g. inverse oflow pass filter) may boost the total treatment depth. Techniques mayalso involve running a revised target controller (e.g. without a cosineeffect) with a low-pass filter, to evaluate the extent to which SA for aclinical data set correlates with observed SA, or to evaluate the extentto which post-operative refractions correlate with what is expectedbased on the clinical data. The Optimized Kernel Parameter can berelated to LPF, and sigma can represent the diffusion coefficient.

Shape Induced SA

FIG. 26 shows a total induced SA (left panel, 0.188±0.139 for myopia and−0.110±0.179 for hyperopia) and a shape-induced SA (right panel,0.064±0.049 for myopia and −0.071±0.038 for hyperopia) after taking intoaccount a low-pass filtering effect, according to embodiments of thepresent invention. When considering the mean, it is possible to observethat shape-induced SA consists of ⅓ of the total SA for myopia and morethan ½ for hyperopia. When considering the trend line slope, it ispossible to observe that shape-induced SA consists of more than ½ formyopia and less than ¼ for hyperopia. Therefore, a shape-induced SA canbe a significant component for an observed post-surgery sphericalaberration. For the data presented in FIG. 26, the healing effect forthe shape-induced SA was included in the simulation.

Low Pass Filter

Assuming that a particular theoretical target shape provides a best fitfor low order correction it is possible to perform an optimization asfollows. First, an ablation target for an eye (e.g. an eye from a study)can be calculated according to a respective scaling factor and sphereadjustment. Second, a low pass filter (e.g. Butterworth or Gaussian) canbe applied to obtain a healed shape. Third, a residual shape can beobtained by subtracting the healed shape from a pre-operative CV(CustomVue®) treatment shape. Fourth, a residual error in SA (e.g.predicted SA) can be calculated. Fifth, a merit function can becalculated. For example, the merit function may be the square root ofthe average sum of the square difference between the observed SA and thepredicted SA. FIG. 27 shows aspects of optimization of a low passfilter, according to embodiments of the present invention. FIGS. 28A and28B show aspects of a kernel and an inverse kernel, according toembodiments of the present invention.

Shape Deconvolution and Verification

According to some embodiments, it is possible to process a target shapeas follows. First, a theoretical target is created, optionally using azone-extended target algorithm. The target shape is then convolved withan inverse kernel. The convolved shape is them lifted to avoid negativeablation. A scaling factor can then be applied to preserved SE over a 4mm zone. Subsequently, a cosine effect can be applied. FIG. 29 depictsaspects of a treatment target deconvolution according to embodiments ofthe present invention.

According to some embodiments, it is possible to verify such targetshape procedures as follows. First, obtain a theoretical target shapefor an eye (e.g. each eye from a study set). Second, obtain adeconvolved target by convolving the target shape with an inversekernel. Third, convolve the target with a determined kernel (e.g. healedtarget). Fourth, calculate the difference between the theoretical targetand the simulated healed target (e.g. healed target subtracted fromtheoretical target). FIG. 30 depicts aspects of a target verificationprocedure according to embodiments of the present invention. FIGS. 31A,31B, and 31C depict residual error with deconvolution, according toembodiments of the present invention.

FIGS. 32A-32C depicts expected targets (left column), inversed convolvedtargets (middle column), and the difference between expected andinversed convolved targets (right column), according to embodiments ofthe present invention.

Optimization of Kernel

FIG. 33 depicts CV data from a study (515 eyes, including myopia,hyperopia, high myopia, and mixed cases, as well as VSSR™ treatment datafrom a Canadian study (77 eyes, including myopia [mostly], and a fewhyperopia and mixed cases). FIG. 33 indicates that the optimized sigmafor various data sets suggests a range between about 0.33 mm and about0.40 mm.

Post-Operative SA (Expected vs. Actual)

FIG. 34 depicts actual vs. expected post-operative sphericalaberrations.

Other Features

FIGS. 35A-35C depicts cylinder like cases, mixed cases, and hyperopiacases, respectively, according to embodiments of the present invention.

Dual Scale Kernel Techniques

Embodiments of the present invention encompass systems and methods whichimplement dual scale kernel techniques, triple scale kernel techniques,and other multi-scale kernel techniques. In some cases, embodiments ofthe present invention encompass multi-scale processes that account forcorneal healing. Exemplary embodiments encompass filters having multipleparameters (e.g. dual scale filters) that can be developed by comparingsimulated and observed data.

Laser vision correction is a rapidly growing field for correctingnearsightedness, farsightedness as well as astigmatism with dominatinglaser-assisted in situ keratomileusis (LASIK) procedures. Suchtechniques works well for correcting spherocylindrical aberrations,although there may be challenges with sufficiently correcting high orderaberrations (HOAs), in particular spherical aberration (SA), due toinduction of HOAs post-surgery. Embodiments of the present inventionencompass systems and methods involving dual-scale linear filteringkernels (and other multi-scale approaches) that model or account forcorneal epithelial remodeling as a source that accounts for HOAinduction processes. As discussed herein, dual-scale kernels weredeveloped based on several retrospective clinical data sets used astraining data sets. In some cases, a downhill simplex algorithm can beused to develop two free parameters of the kernel. The performance ofthe kernel was tested on new clinical data sets that were not previouslyused for the development. Hence, embodiments of the present inventionencompass systems and methods for generating linear filters that predictpost-LASIK corneal smoothing, for example based on training data sets.

Historically, eyeglasses and contact lenses have been used to alleviaterefractive problems such as nearsightedness, farsightedness, andastigmatism. With the advent of excimer lasers specially designed forlaser-assisted in situ keratomeliusis (LASIK) and photorefractivekeratectomy (PRK), patients have started to enjoy a new type of visioncorrection that is free of eyeglasses. With wavefront-guided LASIK, thecorrection of ocular aberrations is no longer limited to the so-calledlow-order aberrations, i.e., the spherocylindrical error that can becorrected with traditional eyeglasses. This technology enables thecorrection of higher-order aberrations (HOAs) that are beyond thespherocylindrical error, most notably spherical aberration and coma.Thus, super sharp vision can be targeted with the wavefront-guidedLASIK.

LASIK often involves first cutting a flap on the corneal stroma, liftingit to the side, then delivering the UV laser pulses to remove tissue,and finally putting back the flap, which heals shortly after surgery.The precise design of an ablation target may cut the corneal stroma asneeded to achieve a desired shape immediately after surgery. However,the biomechanical process and the corneal epithelial remodeling aftersurgery change the surface of the cornea, resulting in deviations fromthe original optical design of the ablation shape. The post-operativeinduction of HOAs, especially spherical aberration (SA), is currentlyamong the most serious challenges for laser vision correctiontechnology. It has been discovered that post-operative cornea remodelingcan be a root cause of SA induction. One main effect of the cornearemodeling is the smoothing of epithelial anterior surface, when theepithelium tends to grow thicker at the center and fill in the dips ofthe cornea surface, created by refractive surgery. The epithelialsmoothing can cause some spherocylindrical regression after refractivesurgery, which can be corrected by a linear adjustment of the intendedrefractive correction. It can also lead to the induction of high-orderaberrations, which can be increasingly strong for high myopia andhyperopia cases.

In general, the amount of the SA induction tends to increase withpost-surgery time. Several months after surgery when the corneastabilizes, the induced SA shows statistically significant trend versusthe magnitude of the treated refraction. FIG. 36 shows thepost-operative SA over a 6 mm diameter as a function of thepre-operative manifest refraction in spherical equivalent (MRSE). Theregression slope of the induction is remarkably consistent betweendifferent data sets. This graph depicts post-LASIK spherical aberration(SA) as a function of the pre-operative manifest refraction in sphericalequivalent (MRSE) for two data sets.

Embodiments of the present invention involve the development of cornealsmoothing techniques that represent the corneal change post-surgery, forexample based on retrospectively available clinical data. Exemplarykernels can be tested with other clinical data sets that were notpreviously used for the development. Such kernels can then be used to“reverse” the biological corneal smoothing effect by a mathematicaldeconvolution process. Vision treatments can be designed so as to reduceor avoid induced spherical aberration.

Various techniques can be used to capture geometric changes to thesurface of the human cornea occurring after the surgery (e.g.post-operative corneal smoothing). For example, a linear filter (LF)technique can be used to describe post-operative smoothing of thecorneal ablation. Such techniques can be characterized by a small set ofparameters determined by a filter development process based onretrospective clinical data. According to some embodiments, thepost-operative epithelial smoothing process can be simulated by atechnique that defines the shape of the post-operative cornea surface asa convolution of the ablation target profile with a linear smoothingfilter as:h _(post-op) =h _(pre-op) −K(x,y)

T(x,y)  Equation 23

where h stands for the elevation maps of the corneal surface forpre-operative and post-operative situations, respectively,

denotes a convolution operation, T(x, y) is the ablation target profileand K(x, y) is the linear smoothing filter kernel. A squared Butterworthfilter of the first order can take a form with the square term of thespatial frequency as:

$\begin{matrix}{{K( {k_{x},k_{y}} )} = \frac{1}{1 + \frac{k_{r}^{2}}{s^{2}}}} & {{Equation}\mspace{14mu} 24}\end{matrix}$

where K(k_(x), k_(y)) is the Fourier transform of K(x, y),k_(r)=√{square root over (k_(x) ²+k_(y) ²)}, and s is a parameterrepresenting the scale of the kernel. The term k_(r) ² can berepresented as k_(x) ²+k_(y) ². Eq. (24) is in the Fourier domain.Embodiments of the present invention also encompass dual-scale andtriple-scale linear filters that may have a somewhat similar shape as asquared Butterworth filter. According to some embodiments, various testsshow that a dual-scale linear filter technique has the advantage of fastconvergence and proper account of biological change of the epithelialtissue. For example, a dual-scale or dual parameter linear filter kernelcan be defined as:

$\begin{matrix}{{K( {x,y} )} = \frac{1}{1 + \lbrack \frac{r}{s_{2}} \rbrack^{2} - \lbrack \frac{r}{s_{4}} \rbrack^{4}}} & {{Equation}\mspace{14mu} 25}\end{matrix}$

where r=√{square root over (x²+y²)} is the radial distance from thecoordinate origin, s₂ and s₄ are two free parameters that can bedetermined. Eq. (25) is in the spatial domain. According to someembodiments, r, s₂ and s₄ all have dimensions in mm. Here, the powerexpansion is in the denominator. According to some embodiments, this canbe considered as an inverse of a power expansion. FIG. 37 shows thecross-section of such a kernel and its power spectrum. As depicted here,the left panel provides a linear scale of the center of the kernel, themiddle panel provides a logarithmic scale of the entire kernel, and theright panel provides a power spectrum of the kernel (e.g. in Fourierdomain). As shown in the middle panel, the profile provides a sharp coreand a wide skirt or wings.

By using Eq. (25) in Eq. (23) using the pre-operative and post-operativewavefront data as well as the treatment targets for various previouslytreated eyes, the two kernel parameters s₂ and s₄ can be obtained byminimizing the difference between the simulated post-operative wavefronterror and the observed post-operative wavefront error. This minimizationis a least-squares type which minimizes the regression slopes of thepost-operative spherical equivalent (SE) and post-operative SA as afunction of the pre-operative SE for all eyes as:

$\begin{matrix}{\sigma^{2} = {\sum\;\lbrack {( \frac{{slopeSE}_{simu} - {slopeSE}_{obs}}{\delta\lbrack {slopeSE}_{obs} \rbrack} )^{2} + ( \frac{{slopeSA}_{simu} - {slopeSA}_{obs}}{\delta\lbrack {slopeSA}_{obs} \rbrack} )^{2}} \rbrack}} & {{Equation}\mspace{14mu} 26}\end{matrix}$

where slopeSE and slopeSA are the regression slopes of thepost-operative SE versus pre-operative SE and post-operative SA versuspre-operative SE, respectively. δ stands for 95% confidence interval ofthe observed slope. Subscript “simu” stands for simulation and subscript“obs” stands for observation, i.e., clinical outcome. According to someembodiments, such a two scale technique can involve a short scale (e.g.that accounts for high order aberrations) and a long scale (e.g. theaccounts for regression of lower order aberrations, or refraction).Hence, the short scale can relate to the SA term of Eq. (26) and thelong scale can relate to the SE term of Eq. (26).

Several different development techniques have been tested. It has beendiscovered that a downhill simplex method works well in the context ofthis approach and the data sets. With four clinical data sets (two partsof Data Set 1, low myopia and high myopia, and Data Sets 3 and 5) usedfor development, it was found that s₂=0.0334 mm and s₄=0.464 mm give theminimum a as defined in Eq. (26). Hence, multi-scale kernel techniques,including such dual scale kernel techniques, can provide multi-parameterapproaches that exhibit a good fit between data sets, which are in linewith clinical observations or outcomes. For example, the s₂ and s₄values of Eq. (25) as discussed herein can provide a good fit withclinical data.

With these two kernel parameters, application of the treatmentparameters for those eyes using Eq. (23) makes it possible to obtain thesimulated clinical outcome, which includes spherocylindrical error(wavefront spherical equivalent, or WSE) and SA. The WSE can be measuredin diopters (D) and the SA can be measured in microns (μm) over a 6 mmdiameter. FIG. 38 shows a comparison between the observed and simulatedpost-operative outcome for two data sets that were used for thedevelopment. Both the post-operative WSE and SA as a function of thepre-operative WRSE are plotted. It can be seen that the regressionslopes of the simulated eyes agree well with those of the observed eyes.This comparison shows simulated and observed post-operative aberrations(WSE and SA) for Data Set 1 (left panels, two subsets, n=390) and DataSet 3 (right panels, n=76).

Once the parameters s₂ and s₄ are determined, the linear filter kernelcan be determined based on Eq. (25). To obtain a new target shape thatis capable of removing the post-operative induction of sphericalaberration, a deconvolution process of Eq. (23) can be employed as:

$\begin{matrix}{T_{new} = {{K_{INV} \otimes T_{current}} = {{F\lbrack \frac{\lbrack {K( {k_{x},k_{y}} )} \rbrack^{*}}{\lbrack {K( {k_{x},k_{y}} )} \rbrack^{2} + {SNR}^{2}} \rbrack} \otimes T_{current}}}} & {{Equation}\mspace{14mu} 27}\end{matrix}$

where F(•) stands for a Fourier transform, * denotes a complexconjugate, T_(current) is the current treatment target with induction ofpost-operative SA, T_(new) is the new target that is expected to removethe post-operative SA, and K_(INV) is the inverse kernel of K(x,y).According to some embodiments, this approach can involve a Wienerfiltering technique. The SNR can be used to prevent noise amplificationand oscillation at the edge. A value of 0.1 can be used for practicalpurposes.

The effect of post-LASIK central corneal thickening caused by epithelialsmoothing can provide at least partial explanation for regression afterrefractive surgery for myopia. Linear filter embodiments as disclosedherein can reflect a central corneal thickening phenomena, which givesbiological support of the kernel.

In one verification approach using new test data, a kernel developedwith Data Sets 1, 3 and 5 was applied to Data Sets 2 and 4, as depictedin FIG. 39. Specifically, this figure provides a comparison of simulatedand observed post-operative aberrations (WSE and SA) for Data Set 2(left panels, n=74) and Data Set 4 (right panels, n=72). As shown here,the regression slopes of the simulated eyes agree well with those of theobserved eyes, even though these data sets were not used for thedevelopment. This result is consistent because the technique is intendedto simulate the post-operative corneal smoothing process, which shouldnot be different for different data sets.

Another verification approach was employed, with reference to the use ofSE and SA as two aberration parameters for development in Eq. (26). Thesame kernel was used to obtain similar regression slopes for thesecondary spherical aberration, which is not a parameter used in thedevelopment, as shown in FIG. 40. Here, post-operative secondary SA isdepicted as a function of the pre-operative SE for simulated andobserved eyes in the Data Set 1 (upper left, n=390), Data Set 2 (upperright, n=74), Data Set 3 (lower left, n=76), and Data Set 4 (lowerright, n=76). All eyes are myopic. Good matches can be seen between theobserved and the simulated slopes. This result is consistent, as theinduction of HOAs from the corneal smoothing can be primarilyrotationally symmetric. Secondary spherical aberration is an importantrotationally symmetric aberration, in addition to sphere and primaryspherical aberrations.

Some regression plots show a constant offset between the simulated andthe observed trend lines. These offsets for post-operative SE or SAtrend are about the same for all pre-operative MRSE values, indicatingthat they do not depend on ablation depth. They may be resulted by thecreation of the LASIK flap. Depending on the choice of microkeratome andindividual surgeon technique, the flap-induced aberrations may differfrom site to site or surgeon to surgeon.

Embodiments of the present invention encompass the use of a smoothingkernel to assess the post-operative induction of spherical aberration.Exemplary embodiments provide satisfactory fitting for the regressionslopes for both post-operative low-order refraction and high orderaberrations simultaneously. Development based on both refraction andspherical aberration can lead to diverged outcomes.

Linear filters as disclosed herein having two free parameters canapproximate a dual-scale smoothing. With reference to the cross-sectionof the kernel in logarithmic scale in FIG. 37, the sharp corecorresponds to the short scale diffusion process and the wide wingscorrespond to the long scale smoothing process. These two separatedprocesses can be linked to the post-operative corneal change inlow-order and high-order aberrations, respectively. Consequently, thelinear filter can be considered to yield a good match for both low-orderaberrations (WSE) and high order aberrations (SA) observed clinically.Furthermore, this match can be extended to different data sets anddifferent aberration types (secondary spherical aberration).

With reference to the inverse kernel K_(INV), as depicted in FIG. 41, itcan be seen that the peak of the power spectrum of the inverse kernelcorresponds to the size of the superficial cells of the epithelium. Asshown here, the cross-section of the inverse kernel of the linear filteris in the left panel (e.g. spatial domain), and the power spectrum ofthe inverse kernel is in the right panel (e.g. Fourier domain). Withoutbeing bound by any particular theory, it is believed that thisrelationship may be related to movement of the epithelial cells,especially the superficial cells, as attributed to the mechanism of thepost-operative corneal smoothing. As the smoothing kernels generallysmooth high curvature areas, the effect of the inverse kernel works inan opposite manner, sharpening areas that have high curvature changes.The link of the peak of the power spectrum of the inverse kernel to thesize of the superficial cells of the epithelium can provide anadditional level of confirmation for the kernel development process.

The epithelium of the eye often can have a depth of about 50 μm,representing five layers of cells. Individual superficial cells may havea height of about 5 μm and a diameter of about 20 μm. As depicted in theright panel of FIG. 41, the power spectrum peak can corresponds to thecell diameter (e.g. at about 20 cycles/mm).

As disclosed herein, it is possible to use certain combinations of datasets to develop a kernel, and certain other combinations of data sets totest or evaluate the kernel. A successful test of the developed kernelcan lend strength to the usefulness of the kernel, which can beimportant when determining whether or how to proceed with costlyclinical trials.

All patent filings, scientific journals, books, treatises, and otherpublications and materials discussed in this application are herebyincorporated by reference for all purposes. A variety of modificationsare possible within the scope of the present invention. A variety ofparameters, variables, factors, and the like can be incorporated intothe exemplary method steps or system modules. While the specificembodiments have been described in some detail, by way of example andfor clarity of understanding, a variety of adaptations, changes, andmodifications will be obvious to those of skill in the art. Although theinvention has been described with specific reference to a wavefrontsystem using lenslets, other suitable wavefront systems that measureangles of light passing through the eye may be employed. For example,systems using the principles of ray tracing aberrometry, tscherningaberrometry, and dynamic skiascopy may be used with the currentinvention. The above systems are available from TRACEY Technologies ofBellaire, Tex., Wavelight of Erlangen, Germany, and Nidek, Inc. ofFremont, Calif., respectively. The invention may also be practiced witha spatially resolved refractometer as described in U.S. Pat. Nos.6,099,125; 6,000,800; and 5,258,791, the full disclosures of which areincorporated herein by reference. Treatments that may benefit from theinvention include intraocular lenses, contact lenses, spectacles andother surgical methods in addition to refractive laser corneal surgery.

Each of the calculations discussed herein may be performed using acomputer or other processor having hardware, software, and/or firmware.The various method steps may be performed by modules, and the modulesmay comprise any of a wide variety of digital and/or analog dataprocessing hardware and/or software arranged to perform the method stepsdescribed herein. The modules optionally comprising data processinghardware adapted to perform one or more of these steps by havingappropriate machine programming code associated therewith, the modulesfor two or more steps (or portions of two or more steps) beingintegrated into a single processor board or separated into differentprocessor boards in any of a wide variety of integrated and/ordistributed processing architectures. These methods and systems willoften employ a tangible media embodying machine-readable code withinstructions for performing the method steps described above. Suitabletangible media may comprise a memory (including a volatile memory and/ora non-volatile memory), a storage media (such as a magnetic recording ona floppy disk, a hard disk, a tape, or the like; on an optical memorysuch as a CD, a CD-R/W, a CD-ROM, a DVD, or the like; or any otherdigital or analog storage media), or the like. While the exemplaryembodiments have been described in some detail, by way of example andfor clarity of understanding, those of skill in the art will recognizethat a variety of modification, adaptations, and changes may beemployed.

While the above provides a full and complete disclosure of exemplaryembodiments of the present invention, various modifications, alternateconstructions and equivalents may be employed as desired. Consequently,although the embodiments have been described in some detail, by way ofexample and for clarity of understanding, a variety of modifications,changes, and adaptations will be obvious to those of skill in the art.Accordingly, the above description and illustrations should not beconstrued as limiting the invention, which can be defined by the claims.

What is claimed is:
 1. A method of determining a laser-ablation visiontreatment plan for an eye of a patient, the method executed by acomputer-based treatment-planning system, the method comprising:receiving, at an input of the treatment-planning system, an originaltarget profile for the eye of the patient, the original target profilebased on a desired cornea shape of the eye after the laser-ablationvision treatment; obtaining a spatial domain kernel filter thatsimulates post-operative corneal epithelial remodeling of the eyeoccurring after the laser-ablation vision treatment, wherein the spatialdomain kernel filter is an inverse of a power expansion; convolving theoriginal target profile with the spatial domain kernel filter to obtaina convolved target profile; and determining the laser-ablation visiontreatment plan based on the convolved target profile.
 2. The method ofclaim 1, wherein the power expansion is in spatial coordinates.
 3. Themethod of claim 1, wherein the power expansion is a multi-parameterpower expansion.
 4. The method of claim 1, wherein the power expansionis a dual parameter power expansion.
 5. The method of claim 1, whereinthe power expansion is a triple parameter power expansion.
 6. The methodof claim 1, wherein the spatial domain kernel filter is based on acomparison between simulated post-operative wavefront error data andobserved post-operative wavefront error data.
 7. The method of claim 1,wherein the spatial domain kernel filter is based on a minimum value ofσ, according to the equation$\sigma^{2} = {\sum\;\lbrack {( \frac{{slopeSE}_{simu} - {slopeSE}_{obs}}{\delta\lbrack {slopeSE}_{obs} \rbrack} )^{2} + ( \frac{{slopeSA}_{simu} - {slopeSA}_{obs}}{\delta\lbrack {slopeSA}_{obs} \rbrack} )^{2}} \rbrack}$wherein slopeSE and slopeSA are regression slopes of post-operative SEversus pre-operative SE and post-operative SA versus pre-operative SE,respectively, and wherein δrepresents a 95% confidence interval of therespective slope.
 8. The method of claim 1, wherein the spatial domainkernel filter is represented as K(x,y), and is characterized by theequation${K( {x,y} )} = \frac{1}{1 + \lbrack \frac{r}{s_{2}} \rbrack^{2} - \lbrack \frac{r}{s_{4}} \rbrack^{4}}$wherein r =√{square root over (x²+y²)} is a radial distance from acoordinate origin, and wherein s₂and s₄ are each free parameters.
 9. Acomputer-based system for determining a laser-ablation vision treatmentplan for an eye of a patient, the system comprising: an input configuredto receive an original target profile for the eye of the patient, theoriginal target profile based on a desired cornea shape of the eye afterthe laser-ablation vision treatment; a convolution module comprising aprocessor and a tangible non-transitory computer readable medium, thecomputer readable medium programmed with a computer application that,when executed by the processor, causes the processor to convolve theoriginal target profile with a spatial domain kernel filter to obtain aconvolved target profile, the spatial domain kernel filter simulatingpost-operative cornea remodeling of the eye occurring after thelaser-ablation vision treatment, wherein the spatial domain kernelfilter is an inverse of a power expansion; and a treatment planningmodule configured to determine the laser-ablation vision treatment planbased on the convolved target profile.
 10. The computer-based system ofclaim 9, wherein the power expansion is in spatial coordinates.
 11. Thecomputer-based system of claim 9, wherein the power expansion is amulti-parameter power expansion.
 12. The computer-based system of claim9, wherein the power expansion is a dual parameter power expansion. 13.The computer-based system of claim 9, wherein the power expansion is atriple parameter power expansion.
 14. The computer-based system of claim9, wherein the spatial domain kernel filter is based on a comparisonbetween simulated post-operative wavefront error data and observedpost-operative wavefront error data.
 15. The computer-based system ofclaim 9, wherein the spatial domain kernel filter is based on a minimumvalue of σ, according to the equation$\sigma^{2} = {\sum\;\lbrack {( \frac{{slopeSE}_{simu} - {slopeSE}_{obs}}{\delta\lbrack {slopeSE}_{obs} \rbrack} )^{2} + ( \frac{{slopeSA}_{simu} - {slopeSA}_{obs}}{\delta\lbrack {slopeSA}_{obs} \rbrack} )^{2}} \rbrack}$wherein slopeSE and slopeSA are regression slopes of post-operative SEversus pre-operative SE and post-operative SA versus pre-operative SE,respectively, and wherein δrepresents a 95% confidence interval of therespective slope.
 16. The computer-based system of claim 9, wherein thespatial domain kernel filter is represented as K(x,y), and ischaracterized by the equation${K( {x,y} )} = \frac{1}{1 + \lbrack \frac{r}{s_{2}} \rbrack^{2} - \lbrack \frac{r}{s_{4}} \rbrack^{4}}$wherein r =√{square root over (x²+y²)} is a radial distance from acoordinate origin, and wherein s₂ and s₄ are each free parameters.
 17. Acomputer program product for determining a laser-ablation visiontreatment plan for an eye of a patient, the computer program productembodied on a non-transitory tangible computer readable medium, thecomputer program product comprising: computer code for receiving anoriginal target profile for the eye of the patient, the original targetprofile based on a desired cornea shape of the eye after thelaser-ablation vision treatment; computer code for convolving theoriginal target profile with a spatial domain kernel filter to obtain aconvolved target profile, the spatial domain kernel filter simulatingpost-operative cornea remodeling of the eye occurring after the laserablation vision treatment, wherein the spatial domain kernel filter isan inverse of a power expansion; and computer code for determining thelaser-ablation vision treatment plan based on the convolved targetprofile.
 18. The computer program product of claim 17, wherein the powerexpansion is in spatial coordinates.
 19. The computer program product ofclaim 17, wherein the power expansion is a multi-parameter powerexpansion.
 20. The computer program product of claim 17, wherein thepower expansion is a dual parameter power expansion.
 21. The computerprogram product of claim 17, wherein the power expansion is a tripleparameter power expansion.
 22. The computer program product of claim 17,wherein the spatial domain kernel filter is based on a minimum value ofσ, according to the equation$\sigma^{2} = {\sum\;\lbrack {( \frac{{slopeSE}_{simu} - {slopeSE}_{obs}}{\delta\lbrack {slopeSE}_{obs} \rbrack} )^{2} + ( \frac{{slopeSA}_{simu} - {slopeSA}_{obs}}{\delta\lbrack {slopeSA}_{obs} \rbrack} )^{2}} \rbrack}$wherein slopeSE and slopeSA are regression slopes of post-operative SEversus pre-operative SE and post-operative SA versus pre-operative SE,respectively, and wherein δrepresents a 95% confidence interval of therespective slope.
 23. The computer program product of claim 17, whereinthe spatial domain kernel filter is represented as K(x,y), and ischaracterized by the equation${K( {x,y} )} = \frac{1}{1 + \lbrack \frac{r}{s_{2}} \rbrack^{2} - \lbrack \frac{r}{s_{4}} \rbrack^{4}}$wherein r =√{square root over (x²+y²)} is a radial distance from acoordinate origin, and wherein s₂and s₄are each free parameters.